Okay, I really keep asking myself after I've slugged my way into controlling the center with pawns... what am I supposed to do about it?

You have the center, you have more space, you have more mobility. You have superior maneuvering possibilities. So, now learn about tactics and middlegame, then you can use the advantages of a superior center.

... unless you used the Stonewall Attack to grab the center

Once I have a good centered position, I like to add some support by any of the flanks... say C or F columns with a rook or the queen. That is a strategy that usually gives me good results: