Prep ingredients: Bring a medium pot of salted water to a boil. Peel and finely chop 1 large clove garlic. Finely chop dill fronds, discarding stems. Trim ends from scallions, then finely chop scallion greens, and thinly slice scallion whites, keeping them separate. Coarsely chop olives, removing any pits. Finely grate Parmesan.
Make pesto: In a small bowl, combine dill, scallion greens, half of the garlic, 3 tablespoons oil, 1⁄4 teaspoon salt, and a few grinds pepper.
Cook pasta: Add pasta to boiling water. Cook until al dente, 10–12 minutes. Reserve 1⁄4 cup pasta water, then drain pasta.
Sauté shrimp & olives: Meanwhile, pat shrimp dry. While pasta cooks, heat 1 tablespoon oil in a large skillet over medium-high. Add shrimp, remaining garlic, scallion whites, olives, and a pinch each of salt and pepper. Cook, stirring frequently, until shrimp are curled and light pink, about 3 minutes.
Add spinach & pasta: Add spinach to skillet; cook until wilted, about 1 minute. Add pasta, dill pesto, and reserved pasta water. Cook, stirring, until liquid has reduced by half, about 1 minute. Remove skillet from heat. Add half of the Parmesan in large pinches to prevent clumping, stirring to combine. Season to taste with salt and pepper.
Serve: Divide pasta between shallow bowls. Sprinkle with remaining Parmesan.