Preheat oven to 400 degrees. Place pine nuts in a single layer on a rimmed baking sheet; toast until golden, about 7 minutes. Shake the pan often while baking to make sure the nuts toast evenly. Let cool; finely chop.
Wash and dry 6 quahog shells; set aside.
Coarsely chop clams; in a large bowl, toss clams with pine nuts, garlic, 1 tablespoon softened butter, basil, lemon juice, vermouth, salt, and pepper; add 1 to 2 tablespoons clam liquid. In a small bowl, mix breadcrumbs with 1 tablespoon melted butter.
Divide clam mixture among reserved shells on a baking sheet; sprinkle with breadcrumbs, and bake until golden brown, about 15 minutes. Serve with lemon wedges.