I want to make enough brown butter for approximately 2 qts. Can I do this all in one batch, or do I have to do it in small batches? Will browned butter solidify overnight in the refrigerator? Can this be done?
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Yes to both questions.
When you're doing it in a large batch, use a whisk to keep it moving and to ensure even browning. And be sure that the heat isn't too high (as you probably already know).
And yes, it solidifies like regular butter and can be melted down again in a pan. I've done it many times at home and restaurants. And a little trick to make stop it from browning further (once it's reached the desire state) is to add a little lemon juice off the heat. It cools it down and slows the browning. Otherwise, just dump it in a bowl immediately when it's down to stop the cooking.
In addition to what was said above, I would use a large pan with a lot of surface area. The milk solids will brown more evenly, and it'll be easier to see what's going on in the pan.
Thank you, both answers are very helpful.