Place a rack in the top third of the oven; heat to 350 degrees F. In a medium bowl, toss the shrimp with 1 tablespoon olive oil, half the grated garlic, and the paprika. Season with salt and pepper and set aside to marinate.
Toast the hazelnuts on a sheet pan in the oven until golden brown, 7 to 10 minutes. Let hazelnuts cool slightly, then crush with the bottom of a heavy pan into smaller pieces. Remove loose hazelnut skins (it’s okay if they all don’t come off), then transfer nuts to a small bowl. Wipe off the baking sheet and heat the oven to broil.
Toss the tomatoes and peppers on the reserved baking sheet with 1 tablespoon olive oil to coat. Season to taste with salt and pepper. Broil for 5 minutes, then add the shrimp and broil until the tomatoes and peppers are blistered and the shrimp have just turned pink, 3 or 4 minutes.
Drizzle 3 tablespoons olive oil as well as the vinegar over the contents of the baking sheet and toss to coat. Sprinkle with the red pepper flakes and remaining garlic and let marinate until the mixture has cooled down slightly. (As the mixture sits, the tomato juices will combine with the oil and vinegar and the peppers and tomatoes will soften further.)
Place the baby greens on a serving platter or individual plates. Spoon the contents of the baking sheet and the oil-vinegar mixture over the greens. Top with hazelnuts and parsley and season to taste with salt and pepper.