Blend eggs, milk, Siracha, flour, salt, pepper, thyme in a blender for 30 seconds.
Place butter, oil and mushrooms in 9- or 10-inch oven-proof pan. Heat oven to 425°F. When oven reaches temperature, swirl pan, cook mushrooms 5 minutes more.
Remove pan from oven and spoon mushrooms from pan, leaving as much butter mixture as you can. Pour egg mixture into hot pan. Bake 15 minutes. Remove pan from oven. Sprinkle mushrooms and goat cheese over pancake; return to oven and bake 5 minutes more. Cut into large wedges to serve.