Preheat the oven to 400 degrees F.
Use a bread knife to cut a cross-hatch pattern into the top of the bread, stopping short of cutting through to the bottom of the loaf (so the loaf holds together).
Melt the butter in a small skillet over low heat. Add the olive oil and garlic and cook, stirring often, until very pale golden, about 4 minutes. Stir in 3/4 teaspoon salt and the parsley. Remove from the heat.
Gently separate the cubes of bread from the top of the loaf. Spoon some of the garlic butter in the cuts and spread it over the cut surfaces; use all but 2 tablespoons of the garlic butter for this. Slather the remaining garlic butter on top of the loaf. Distribute the Asiago between the cuts and on top. Wrap the loaf in foil and place it on a baking sheet.
Bake the bread until heated through, about 10 minutes. Unwrap the foil and continue baking until slightly crisp on the outside, 5 to 7 minutes more. Serve immediately.