ROAST THE BRUSSELS SPROUTS: Preheat oven to 400 degrees F. In a large bowl, toss Brussels sprouts with 3 tablespoons of the olive oil along with 1 teaspoon of the salt, ½ teaspoon of the pepper, and the ground cayenne (you’ll use the rest of the oil and salt and pepper with the mushrooms and onions later.) Turn them out onto a rimmed baking sheet and roast in the oven for 30-35 minutes, stirring half-way through, or until tender and browned. Remove from the oven, leaving the oven on.
SAUTE THE MUSHROOMS, ONION, AND GARLIC: Meanwhile, in a large deep skillet or saute pan, heat the butter along with the remaining tablespoon of olive oil over medium heat. Tip in the mushrooms and onion and sprinkle with the remaining 1/4 teaspoon salt and 1/4 teaspooon pepper or to taste. Cook and stir for about 10 minutes or until quite tender, adding the garlic for only the last minute or two.
ADD BRUSSELS SPROUTS TO MUSHROOM MIXTURE, ADD CHEESE/RETURN TO OVEN: When both the Brussels sprouts and the mushroom mixture are done, add the Brussels sprouts to the large skillet with the mushroom mixture and stir gently, but well. Sprinkle evenly with the blue cheese and return to the oven for 7-8 minutes or just until the cheese is well-melted. Serve hot.