In summer, this bread can be cooked on the grill; in cold weather, it’s just as good prepared in the oven.
Blend butter and rosemary in processor. Gradually add oil and process until incorporated. Mix in Parmesan cheese. (Can be made 1 day ahead. Cover and refrigerate. Bring to room temperature before continuing.) Mix in garlic.
Prepare barbecue (medium-high heat) or preheat oven to 400°F. Spread cheese mixture over cut sides of bread.
If grilling, wrap each piece of bread loosely in foil. Grill in foil until bread is crusty and golden, about 8 minutes. If baking, place bread cut side up on baking sheet. Bake until bread is crisp and golden around edges, about 10 minutes. Serve bread warm.