Combine the balsamic vinegar, brown sugar, and 1 Tbs. water in a small dish and set near the stove. Set a shallow serving dish near the stove, too. Turn on the exhaust fan. In a 10-inch straight-sided sauté pan, heat 1 Tbs. of the butter with the olive oil over medium-high heat. When the butter is melted, add the mushrooms and 3/4 tsp. salt and stir right away with a wooden spoon until the mushrooms have absorbed all the fat. Let the mushrooms cook undisturbed for 2 minutes and then stir once. The pan will look dry, but keep the heat at medium high and continue to cook, stirring infrequently, until the mushrooms are shrunken, glistening, and some sides have developed a deep orange-brown color, 6 to 7 minutes more (the bottom of the pan will be brown). Turn the heat to low, add the garlic and the remaining 1 Tbs. butter and cook, stirring, until the butter is melted and the garlic is fragrant, 15 to 20 seconds. Carefully add the balsamic mixture (you’ll need to scrape the brown sugar into the pan with a rubber spatula). Cook, stirring, until the liquid reduces to a glazey consistency that coats the mushrooms, 15 to 20 seconds. Season with a few grinds of pepper. Immediately transfer the mushrooms to the serving dish, scraping the pan with a rubber spatula to get all of the garlicky sauce. Let sit for a few minutes and then serve warm.