Should you cover the pot when simmering?

Hi Brian! Thanks for your great question. +1 for what @boulangere offered below -- I'd suggest adding 3-4 quarts of water. And no need to cover the pot while boiling/simmering (though it does happen quicker!).

Start with a gallon of water. In culinary school I was taught to bring a covered pot to the boil, because a covered pot boils faster than an uncovered one, then remove the lid, reduce heat to a simmer, and cook down until desired flavor is reached. I'm pretty sure I could recite that in my sleep, and perhaps have. That said, in the past few years, I've tried to be sure I allow enough cooking time that I can leave a pot partially covered so that it reduces more slowly. I find that the resulting stock has a much better concentration of flavor, and also a lovely deep color.