Lay painter's tape around the border of the wood that's going to be varnished. Put newspaper on the floor below it.
Wipe down the stained woodwork with mineral spirits, getting it completely clean. Let it dry. Rewipe it with a tack cloth to take up any lingering dust.
Mix a 50-50 "sealing'' solution of varnish and mineral spirits in a small bucket. Stir gently with your foam brush to thoroughly mix it, taking care not to form bubbles in the mixture.
Brush the 50-50 varnish solution on the wood, going with the direction of the grain and starting at the top of the area. Make the coat of solution very thin and even. Let it dry overnight.
Lightly buff the dried sealing solution with 320-grit sandpaper, just enough to remove the shine. Clean off the dust with a tack cloth.
Reapply a second layer of sealing solution in the same manner as the first. Let it dry overnight. Buff it as before, and wipe off the dust with a tack cloth.
Mix a solution of 4 parts varnish, 1 part mineral spirits. Apply a coat of the thinned varnish in the same manner as before, keeping it smooth and even with no bubbles. Let it dry overnight.
Buff the varnish, wipe off the dust and repeat the process for at least three coats. Don't buff the final coat, but let it cure for at least two days before resuming use of the area.