1 head fresh garlic
2 1/2 tablespoons extra-virgin olive oil plus more for shrimp and grill
4 cups shelled peas (from 4 to 4 1/2 pounds in pods) or 2 (10-ounce) packages frozen peas, not thawed
2 tablespoons unsalted butter
Fine sea salt
Freshly ground black pepper
3 sprigs fresh thyme
20 jumbo shrimp, preferably with heads
Heat oven to 375° with rack in middle. Cut off and discard top 1/2 inch of garlic head to expose cloves. Set head on a sheet of foil, cut-side up, and drizzle with ½ tablespoon oil. Wrap garlic in foil and roast 1 hour, or until very tender.
While garlic is roasting, in a medium saucepan, combine 3/4 cup water, peas, butter, and 1/4 teaspoon salt. Bring to a simmer and cook until peas are tender, about 5 minutes. Reserving cooking liquid, strain peas. Let peas cool.
Place half of the peas in the bowl of a food processor. Add 2 tablespoons cooking liquid, then purée until smooth. Transfer purée to a large bowl.
When garlic is very tender, remove from oven and let cool. Squeeze roasted garlic cloves into bowl with pea purée; discard skins. Add remaining 2 tablespoons oil and generous pinch salt and pepper; whisk purée to combine. Adjust seasoning to taste.
Prepare a charcoal grill for direct-heat cooking over medium-hot charcoal (medium-high heat for gas).
Remove leaves from 1 thyme sprig.
Peel shrimp, leaving heads on. Drizzle shrimp with oil and season with thyme leaves, salt and pepper, then grill on lightly oiled grill rack, turning occasionally, until cooked through, about 5 minutes.
Spread pea purée in center of serving plates. Top with shrimp and remaining peas. Sprinkle with pepper and garnish with remaining thyme sprigs.