Bring a large pot of salted water to a boil.
Add the cheese tortellini and roughly chopped broccoli florets to the boiling water and cook according to the tortellini package directions until the tortellini is al dente, about 4–6 minutes.
During the last minute of cooking, add the thin lemon slices to the pot.
Just before draining, reserve 1 cup of the pasta cooking water, then drain the tortellini, broccoli, and lemon slices in a colander and pick out and set aside the lemon slices.
Return the empty pot to medium heat and add the olive oil and the reserved lemon slices; fry the lemon slices until lightly golden on each side, about 30 seconds to 1 minute.
Remove the pot from the heat and add the drained tortellini, broccoli, baby spinach, chopped chives, butter, grated manchego, and grated parmesan.
Toss everything together, adding a splash of the reserved pasta water at a time until the cheeses melt and a loose sauce forms; add more pasta water as needed.
Season to taste with crushed red pepper flakes, kosher salt, and black pepper, then serve.