Bring a large pot of salted water to boil, and cook pasta according to package directions.
While the pasta is cooking, chop onions, sage, and broccoli into bite-sized pieces. Julienne the carrots. Set aside.
Melt 4 tablespoons butter with olive oil in a large skillet over medium heat. Add carrots and broccoli and cook until slightly soft, 5 to 6 minutes. Add the sage and onions and cook, stirring, until lightly brown, about 7 to 8 minutes.
Melt the remaining 2 tablespoons butter and mix with the lemon juice and salt and pepper to taste. Set aside.
Drain the pasta and return to the pot. Mix in vegetables and then toss with the lemon-butter mixture. Mix in Parmesan and serve immediately.
