Preheat oven to 400°F.
Place a large skillet over medium-high heat with the butter and EVOO. Once the butter has melted, add the onions, garlic and thyme and cook, stirring frequently, for about 20 minutes, until the onions are brown and caramelized. If the edges start to get too much color, just add a little water and stir it up.
While onions are caramelizing, score the loaf parallel and perpendicularly to form one inch cubes, making sure not to cut through the base of the loaf.
Arrange loaf on a baking sheet and carefully loosen to form small crevices between the cubes. Stuff the caramelized onions and cheese in between the cubes, making sure it is evenly distributed throughout the whole loaf. Bake until cheese is melted and browning, about 10-12 minutes.