Prepare your vegetables: Finely chop yellow onion, cube fresh tomatoes and squash. In a large stock pot, heat up olive oil on medium until the pot is warm to the touch. If your oil starts to smoke, it's too hot. When the oil is hot, add the garlic and onion, stirring frequently until they become transparent.
Next, add the squash (NOT the tomatoes) and saute together for approximately 5 minutes.
Add in the can of tomato paste, mixing together over heat until the sauce becomes very fragrant. Then, add in the crushed tomatoes, white sugar, balsamic vinegar, salt, oregano, and cayenne pepper. Bring to a boil for 2 minutes then turn heat to low. Let simmer for 20 minutes, stirring frequently.
While the sauce is simmering, boil a pot of water for the bucatini, cook according to instructions, then drain, keeping 1/2 cup of the pasta water, pouring it into the simmering sauce.
When the tomato sauce has finished its 20 minutes, take it off the heat and stir in the fresh, diced tomatoes. Place the bucatini noodles in a serving dish, and coat with sauce to your preference. Top with chopped basil and parmesan cheese, serve immediately.