Place butternut, cut-side up, on a baking sheet. Brush cut sides with about 1 tablespoon olive oil; season with salt and pepper.
Bake in the preheated oven until flesh is tender when punctured with a fork, about 45 minutes. Remove from oven and let squash cool on baking sheet; remove peel and slice into 1/2-inch thick by 2-inch long pieces.
Whisk vinegar, mustard, basil, garlic, brown sugar, salt, and black pepper together in a bowl. Slowly drizzle oil into vinegar mixture while constantly whisking until dressing is silky and emulsified.
Bring a pot of salted water to a boil; cook shrimp until they are bright pink on the outside and the meat is no longer transparent in the center, 4 to 5 minutes. Drain shrimp and sprinkle with lemon juice, salt, and pepper.
Divide lettuce between 2 serving plates; top with butternut squash, tomatoes, and onion. Generously brush dressing onto squash and tomatoes. Arrange shrimp evenly on both plates.