According to On Food and Cooking, mushrooms are about 80-90% water. It makes sense, then, that when frying the mushrooms I should aim to remove as much of that water as possible. That way, I could actually start "frying" and could begin caramelisation and the Maillard Reaction. Bon Appetit give that exact advice here:  If you keep the heat low, the mushrooms will just simmer in their liquid. Medium high or high heat will get rid of all that liquid, and will give the mushrooms a nice brown color.  Advice from On Food and Cooking, however, is the exact opposite:  Their flavor is generally most developed and intense when they are cooked slowly with dry heat to allow enzymes some time to work before being inactivated, and to cook out some of their abundant water and concentrate the amino acids, sugars, and aromas.  Which approach will work best for extracting the most flavour from mushrooms: low heat, or high heat? And why? 
I am with Bon Appetit on this one, or even more extreme - I always pan-fry mushrooms on the highest heat setting. For me, the keys for nice, browned mushrooms on a domestic stove are:  Use very high heat, and preheat the pan before the first batch. Don't crowd the pan, make a single layer of mushrooms. Use a proper turning/stirring regime. That mostly includes to have the patience to wait for the mushrooms to brown well on one side before flipping them to another.  McGee's advice sounds intriguing, but note that he specifies  with dry heat  At least for me, that is incompatible with the "cooked slowly" suggestion. Whenever I try to pan fry mushrooms on medium or low heat, the water they exude stays around, instead of instantly evaporating the way it does on a hot pan. They just stew around in their own sauce, which makes them soggy, bland, rubbery, and prevents caramelization. I don't know why it works for McGee; it might be that he has some technique trick I don't know, or maybe he simply leaves mushrooms in a low oven overnight with the intention to produce a kind of undiluted stew, or even to process them further to something marmitey. But for pan-frying, I have never had success with medium or low temperatures. 