Add the broccoli florets to a large pot of boiling, salted water and cook until al dente, 3 to 4 minutes. Prepare a bowl of ice water. Remove the florets from the pot, reserving the cooking water. Stop the cooking process by shocking them in the ice water. Remove the florets and pat dry.
Add the spaghetti in the same pot of water. Cook until al dente, about 8 minutes.
Meanwhile, add the oil, sliced broccoli stems, red pepper flakes and garlic to a large skillet over medium-low heat and sprinkle with salt. Cook until the garlic is lightly toasted and fragrant, about 8 minutes. Add the florets, toss to coat and warm through.
Drain the pasta in a colander set in the sink, and add the pasta to the skillet. Increase the heat to medium-high. Toss the ingredients and adjust the seasoning if needed. Transfer the pasta to a large serving bowl. Squeeze the lemon juice over the top and sprinkle with the Parmigiano-Reggiano. Serve.