I'll write up another post in a bit with suggestions on how to handle the specific problem you mentioned.
The first rule of leadership in a small office setting (3-10 subordinates) is that you want to be constantly training and cross-training your subordinates so that each person can do the work of two or more positions. You want to cross-train people so that if any one or two are sick or quit, their co-workers can step smoothly and instantly into their slot. (And that includes your own work: You want to make yourself redundant. You want to be able to take a long vacation and be sure that the office will continue to function normally without you.)
Because of all the training and cross-training that's occurring, frequently a project goes to someone who isn't the best candidate for that particular assignment. But that's fine. It becomes a training (or cross-training) opportunity. Don't always give the sensitive assignments to the same one or two people all the time; mix it up and give someone who is weak at a particular skill an opportunity to strengthen that skill.
Maybe your number 2 guy isn't necessarily the best self-starter or independent worker? (Perhaps he was promoted because of his technical skills rather than his ability to work independently.) So do you bypass your number 2 guy and give the desirable independent assignment to your number 3 guy or number 4 guy? Hell no. The number 2 guy will rebel, and rightfully so--he has earned that assignment. So you assign the desirable independent job to the number 2 guy and you treat it as a training (or cross-training) opportunity. If your number 2 guy is weak on self-starter skills, then you work with him and make him stronger on those skills.
Okay, so let's get back to the situation in the thread. You have assigned an independent assignment to someone who turns out to be unable to make the simplest decision by themselves. Which decisions should you make for them? How much access do you want to give them to you and your time? How closely should you watch their work? Obviously much depends on the sensitivity of the assignment.
The general rule is that you meet with them periodically to discuss progress and compare notes as to what they've done and where they're going next. You may decide to meet when they progress from one stage of the project to the next. Or you may decide to meet once a day in the morning or late in the day. Or you may want to meet specifically when certain thorny issues come up. Whatever works best depending on the nature of the project.
However, at the same time you want to encourage maximum independence on the part of the subordinate, if only so that you're not spending all day meeting with subordinates and making all their decisions for them. So a good rule is to tell them: "Don't bring me problems; bring me solutions." In other words, if subordinates come to you with a problem, they should also have researched and be prepared to recommend what they believe to be the best solution for that problem (and perhaps be prepared to offer a plan B in case you don't like plan A).
That way you can either approve their recommendation or you can tell them that you have some concerns with their plan and send them back to come up with a new recommendation that takes care of your concerns.
The point is that they are supposed to do the footwork and research on their own and come up with solutions. You double-check their work and either approve their recommendation or you send them back to try again. In the process, they get exposed to your way of approaching problems (and vice versa), and over time they get better at anticipating how you want problems handled. And in turn, you gain more confidence in how they handle decisions, and you give them more independence to handle assignments without supervision.
But always remember that a supervisor's job is to supervise and train. Don't just throw an assignment at them and walk away and leave them clueless as to what they're supposed to do. Give them as much support and access to you as they need; dig up some paperwork for similar projects in the past that they can use as a guide; be available for questions. But at the same time, push them to come up with solutions on their own and then double-check their decision-making process to ensure that it encompassed all the necessary variables: Tell them: "Don't bring me problems; bring me solutions."
As a final note: Don't wait for special assignments or special circumstances before training people. Training should be an ongoing, daily occurrence. For example: On a daily level, try not to do any work yourself. Assign projects to people below you and have them do the work on their own; then look at the work, correct it, and give it back to them to be redone. Repeat until they get it right. Give them as much support and access to you as they need; but insist that *they* do the work (and that they keep re-doing it) until they get it right.
Summing up: A supervisor's primary job is to monitor and train. If you're doing the office gruntwork because you don't trust your subordinates to do it right, then you're not doing your job. Give it to your subordinates, and then monitor and train them until you're comfortable that they can function independently. Then cross-train them to do each other's jobs as well.