how to melt and cool butter for wet ingredient mixture for muffins?

You heat it til it melts then you leave it on the counter until it is no longer hot

Thanks guys!!

I agree with drshakyhands. I'll add that I often mix the milk in with the butter before adding it to the eggs so the milk can cool the butter down. This way I don't have to worry about if I cooled the butter long enough and I don't risk cooking the eggs accidentally.

What I usually do is gently heat the butter in the microwave until most of it is melted, usually about 20-30 seconds, depending on how much I have to use. I aim to still have a few chunks in there. Then I swirl the container I melted the butter in to get the rest of it to dissolve. I always try to do this step first (hence you must read the entire recipe first too!) and by the time I have everything else together, it's cooled enough to not cook the eggs or milk that you may be adding. No more science to it than that.

I have also seen some recommendations to cut your butter into small pieces and place on top of your oven as you are preheating it. By the time you oven is pre-heated (for some temperatures this may take my oven as long as 20-30 minutes), the butter is pretty soft and nearly melted. I think that "melted butter" implies liquid, versus "softened".