Preheat oven to 400 degrees. In a large nonstick skillet, melt 1 tablespoon butter over medium. Add shallots, garlic, and thyme and cook, stirring, until shallots are translucent, 4 minutes. Add mushrooms and season with salt and pepper; cook until mushrooms are tender and any liquid has evaporated, 20 minutes. Let cool 5 minutes, then stir in goat cheese.
Lay 1 sheet phyllo on a work surface and lightly brush with melted butter. (Keep extra phyllo covered while you work.) Stack 3 more sheets on top, brushing each with butter. Cut stack into 12 squares. Repeat with remaining 4 sheets and more butter to make 24 squares total.
Top each square with 1 tablespoon mushroom mixture. Gather and pinch phyllo around mushroom mixture to form purses with ruffled tops. Bake on a parchment-lined baking sheet until golden brown, about 12 minutes. Let cool slightly before serving.