Trim, stem, and thinly slice mushrooms so all pieces are consistent in size. Wash mushroom pieces in a bowl of cold water; strain.
Heat a large Dutch oven or heavy-bottomed skillet over medium-high heat until hot. Add mushrooms (don’t worry about crowding) and cook, stirring every 2 minutes, until they release their liquid and the pan is dry, about 12–14 minutes. Add olive oil and toss mushrooms; cook until browned, about 3 minutes.
Push mushrooms to sides of skillet, then add butter to the center. When butter has melted, add garlic and ginger, and cook, stirring constantly, until aromatic, about 30 seconds. Add soy sauce, mirin, and vegetable broth as needed, scraping up any browned bits on the bottom of the pan, and stir the mixture into the mushrooms. Cook, stirring occasionally, until liquid has evaporated, 2–3 minutes. Remove from heat and toss with sesame oil.
Garnish with chives, sesame seeds, and a few grinds of pepper; serve warm.