Bring a large pot of salted water to a boil. Add the broccoli florets and cook for 3 minutes, or just until slightly tender. Using a slotted spoon, transfer the cooked broccoli to a colander in the sink. Leave the cooking water in the pot. Rinse the broccoli with cold water to stop the cooking process. Pat the florets dry and set them aside.
Into the same pot of water used for the broccoli add the spaghetti. Cook according to package directions for al dente consistency. Drain the pasta and set it aside.
Into a large, deep skillet over medium-low heat, add the olive oil and butter. When the butter is melted, add the sliced broccoli stems. Cook for 3 minutes, until slightly tender. Add the garlic, salt, and pepper and continue cooking for an additional 30 seconds.
Stir in the broccoli florets, tossing to coat.
Increase the heat to medium-high. Add the spaghetti and cook for 3 minutes to heat through, tossing to coat. Remove from heat, stir in Parmesan cheese, and serve hot.