Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Add shrimps; cook until opaque, 2 to 3 minutes per side. Transfer to a bowl.
Cut zucchini into noodles using a spiralizer fitted with the large shredding blade. Chop up into slightly shorter pieces.
Heat 1 tablespoon olive oil in the same skillet over medium heat. Add zucchini noodles, garlic, 1/2 teaspoon sea salt, and 1/4 teaspoon pepper. Cook and stir until slightly softened, 2 to 3 minutes. Add lemon zest, spinach, and chicken broth. Cook until liquid reduces by 1/4, about 5 minutes. Stir in the shrimp and capers. Cook until shrimp is warmed through, about 1 minute more. Drizzle the remaining olive oil on top.