I've looked at other answers to this question but still have a problem. Application of multiple thin coats is always recommended but the only way I can get a thin coat uniform over a large piece is by "brushing over" when moving from one patch of varnish to the next. The result is thin but dries rough. Isn't there an equally durable varnish that dries slowly enough that brushing over won't result in the roughness? Or is my idea of a thin coat too thin?
It is not unusual to have a "rough" coat as perceived by the user, happens to all. You should make sure to fine sand your base before you urethane, make sure you use a tack rag to clear the surface of bits that will can tribute to this. A tack rag is like a piece of gauze with sticky wax on it. You can thin the urethane with mineral spirits, a favorite mixture of many wood workers. Apply 1 smooth stroke with the brush, do not go over the same area again, you will just disturb and rough up the urethane on first stroke. Remember this is a multi layer process, multi step process.
Fine finisher sand in between coats, especially the first, that's were most the bumps come in due to surface dust, use 200-400 grit at first lightly and then go up to 600-800. Actual grits could get higher depending on your roughness.
Summary. Sand surface well to high grit, clean with tack rag. Apply thin or normal coat. Sand, use tack rag again, apply, sand, tack rag, apply.
There are a few ways to do this. If you want to use a brush, apply it fairly heavily and quickly then let it settle. It should self level if there is enough. You could also add a bit of silicone oil to it to reduce the surface tension and get a smoother finish. Be aware if you do that you will never be able to use a finish that doesn't have silicone oil added to it.
Another way to get it nice an smooth is to do a rubbed finish. Instead of applying with a brush, use a rag or wool applicator. This will allow you to do multiple things coats and get a nice smooth finish. You can also use the silicone oil with this with the same caveat. To get a durable finish you will need 3 or 4 coats at a minimum.
Finally your last and most expensive option is using a sprayer. Either hplv or airless will work. Just be careful when selecting one, you do get what you pay for. Though if you do decide to go this route hplv is probably going to be less expensive.