Make the butter: In a food processor, add the butter, garlic, lemon zest and juice, scallions, parsley, Italian seasoning, a nice pinch of salt and a few grinds of black pepper. Blend all until smooth, then taste and season with more salt if needed.
Make the breadcrumbs: In a small bowl, mix the breadcrumbs, cheese and parsley.
Assemble and grill: Preheat a grill to 600 degrees F.
Arrange the oysters in their half shells on the back of an upside-down baking rack. Add a heaping teaspoon of the butter to each. Sprinkle each with a bit of the breadcrumb mixture. Place the rack on the grill over the direct heat and cook until the oysters are firm and the breadcrumbs are lightly golden, 6 to 8 minutes. Serve with lemon wedges for spritzing.