Bring a large pot of salted water to a rolling boil.
Add the cheese tortellini and chopped broccoli to the boiling water and cook according to the tortellini package until the tortellini is al dente, about 4–6 minutes; add the lemon slices during the last minute of cooking.
Just before draining, reserve 1 cup of the pasta cooking water, then drain the tortellini, broccoli, and lemon slices in a colander.
Return the empty pot to medium heat and add the olive oil and the cooked lemon slices; fry the lemon slices until lightly golden on each side, about 30 seconds to 1 minute per side.
Remove the pot from the heat and add the drained tortellini, broccoli, baby spinach, chopped chives, butter, Manchego, and Parmesan.
Toss everything together, adding a splash of the reserved pasta water at a time until the cheeses melt and form a loose sauce; add more water as needed to reach desired consistency.
Season the pasta with crushed red pepper flakes, kosher salt, and freshly ground black pepper to taste, then serve.