Place garlic, cut side up, onto large sheet of aluminum foil. Drizzle with olive oil and sprinkle with ¼ teaspoon salt and ¼ teaspoon pepper. Tightly wrap heads in foil. Bake until garlic is deep golden and soft, 50 minutes to 1 hour.
When cool enough to handle, squeeze garlic into a medium bowl, discarding skins. Using a fork, mash cloves into a paste. Fold in butter, Italian seasoning, remaining ½ teaspoon salt, and remaining ¼ teaspoon pepper. Place in a covered container until ready to use. Compound butter can be made up to 3 days in advance and refrigerated; let come to room temperature before continuing to next step.
Preheat a grill to medium-high heat (350° to 400°), or heat a cast-iron grill pan on medium-high heat. Brush grill or grill pan with canola oil.
Spread 1 teaspoon compound butter on each side of bread. Grill until lightly golden, about 2 minutes. Turn each slice over, sprinkle with cheese and parsley, and grill until golden, about 1 minute more. Serve immediately with extra compound butter. Garnish with parsley and cheese, if desired.
Notes
If grilling bread in an alfresco setting, make compound butter and slice bread ahead of time, and transport in airtight containers. For garnish, use a Microplane to achieve pretty shavings of Parmesan, if desired.
Recipe by Victoria at https://victoriamag.com/grilled-garlic-bread/