Wash and clean the sardines by carefully rinsing the insides under running water. Remove the heads if desired.
Place the cleaned sardines in a flat dish. Drizzle with olive oil and squeeze lemon juice over them. Season with freshly ground black pepper, paprika powder, and 2 tablespoons thyme. Skin the garlic cloves and cut into thin slices, then scatter them over the fish with a small sprig of rosemary on each. Cover with aluminum foil and refrigerate for at least 1 hour.
Preheat a charcoal grill to medium-high heat.
Remove the sardines from the dish and place them on a cutting board. Stuff each sardine with roughly 1 teaspoon of herb butter and 1 basil leaf in the belly. Season lightly with sea salt. Place the sardines on the hot grill and cook for 3 to 4 minutes per side, turning gently, until the skin is crispy.