Bring a large pot of salted water to a boil.
Add the tortellini and chopped broccoli to the boiling water and cook until the tortellini is al dente, about 4–6 minutes (follow package timing).
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.
Return the empty pot (or heat a large skillet) over 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, then remove the pot or skillet from the heat.
Add the drained tortellini, broccoli, baby spinach, chopped chives, butter, grated manchego, and grated parmesan to the pot with the fried lemon.
Toss everything together while adding a splash of the reserved pasta water, adding more a little at a time until the cheeses melt and form a loose sauce.
Season with a pinch of crushed red pepper flakes and kosher salt and freshly ground black pepper to taste, then toss again and serve.
