Cook pasta according to package directions. Reserve 1/2 cup pasta water before draining.
Meanwhile, in a very large skillet or pan, heat 2 tablespoons oil over medium heat. Add broccoli and stir to coat with oil. If using diced carrots, add them at this time. (If using julienned carrots, wait and add them with onions.)
Sauté until broccoli is very bright green and slightly tender (about 8 to 10 minutes).
Push vegetables to one side of the pan, add remaining oil, and stir in onions until they are coated with oil. If using julienned carrots, add them at this time.
Mix together broccoli, carrots, and onion and continue sautéing until onions begin to appear translucent. (This will take about 7 minutes, but time will vary depending on the size of your pan and the heat of your stove.)Stir often to prevent vegetables from scorching.
Add minced garlic and sesame seeds. Sauté until garlic is cooked but not browned (about 1 minute).
Stir in seasonings, lemon juice, mayonnaise, and reserved 1/2 cup pasta water.
Combine cooked and drained pasta with vegetables and serve immediately.
Notes
Be sure to dice broccoli stems very small (about 1/4-inch) so they cook as fast as the florets (3/4-inch). I use my julienne peeler to cut the carrots. It's such a cool little tool! It only takes a minute to julienne a large carrot and this size of carrots works perfectly in this dish.