Bring a large pot of salted water to a boil and add the broccoli florets; cook 2 to 3 minutes until bright green and slightly tender.
Transfer the broccoli immediately to an ice bath to stop cooking, then drain thoroughly.
Place the drained broccoli in a food processor with a big pinch of kosher salt and black pepper, the minced garlic, baby arugula, 1/2 cup grated Parmesan, lemon juice, and olive oil; blend until smooth and sauce-like, scraping down the sides as needed. Taste and adjust seasoning.
Bring another pot of salted water to a boil and cook the tortellini according to package directions (about 3–5 minutes); drain the tortellini and transfer to a large bowl.
Add a few scoops of the broccoli pesto to the warm tortellini and gently toss to coat, adding more pesto until the pasta is evenly coated.
Fold in the halved cherry tomatoes and the shaved Parmesan; toss gently and serve warm or at room temperature.
