Line a large, rimmed baking pan with parchment paper. Mound tablespoons of the cheese on the parchment and spread each mound to make a thin, even layer, about 3 inches wide. Bake in the oven until medium golden brown all over, 3 1/2 to 5 minutes. Keep a close eye on them, they can burn quickly. Cool completely and transfer to a plate (they will crisp as they cool).
Toss the squash with 2 tablespoons of the oil, 1/2 teaspoon salt, and a little pepper on the baking pan. Bake in the oven until tender and golden brown, about 15 minutes.
Meanwhile, add the shrimp and 2 tablespoons of the pesto to a medium bowl. Stir to coat the shrimp in the pesto. Heat 2 tablespoons of the oil in a large straight-sided skillet over medium-high heat. Add the shrimp to the pan and cook, about 2 minutes, then flip the shrimp and continue cooking until the shrimp are pink and cooked through, another 2 to 3 minutes. Remove the shrimp to a plate and set aside.
Stir the champagne vinegar, the remaining 1/4 cup pesto, and the remaining tablespoon oil together in a small bowl.
Toss the arugula in just enough dressing to coat. Transfer to a large, wide platter and top with the squash, pomegranate seeds, and shrimp; drizzle with more dressing. Break the frico up a little bit, sprinkle on top, and serve immediately.