Peel the onions and cut into thin slices. Place the oil and onion in a large skillet and cook over medium heat for 20 to 25 minutes, or until the onions are golden brown. Remove from heat and set aside.
While the onions are cooking, cut the florets from the broccoli and discard the stems. Bring a large pot of salted water to a boil and add the pasta. Cook over medium-high heat, stirring occasionally for 8 minutes, or until al dente. After the pasta has been cooking for 4 minutes, add the broccoli florets to the pasta water for the remainder of the cooking time. Drain the pasta and broccoli in the colander.
Add the drained pasta and broccoli, almonds, and balsamic vinegar to the pan of onions and mix well. Season with salt and pepper and serve immediately.