Cook the pasta in boiling, salted water for 4 to 6 minutes, or until just tender. Drain the pasta and toss it with 1 tablespoon of the oil to keep it from sticking together, then keep it warm.
Heat the remaining 2 tablespoons of olive oil in a skillet. Add the garlic and saute it lightly; do not brown. Add the broccoli flowerets and peppers, and saute over medium heat until tender, about 5 to 7 minutes. Remove the vegetables from the skillet.
Add the cream to the pan and bring to a boil. Reduce the heat and simmer until it starts to thicken, about 5 minutes. Add the Parmesan cheese and nutmeg, and cook an additional 2 minutes to thicken it a bit more. Add vegetables and the cooked pasta. Toss them together well. Season with salt and pepper if needed. Serve immediately, with additional Parmesan on the side.