Heat olive oil in a 4 to 5 quart pot on medium-high heat.
Add onions and cook for 2 minutes.
Add garlic and cook for 30 seconds longer or until fragrant.
Add the carrots and mushrooms, and cook for 2 minutes.
Add the tomato paste, stir to blend, and cook for a minute longer.
Add the canned tomatoes, basil, and parsley.
Mash and stir the tomatoes with a potato masher until the sauce starts to thicken, about 5 minutes over medium-high heat.
Reduce heat to low and simmer gently while preparing meatballs, stirring occasionally.
In a large bowl, mix by hand the beef, Italian sausage, basil, parsley, mushrooms, eggs, breadcrumbs, cheese, and 2 teaspoons of salt and pepper.
Do not overmix or the meatballs will be tough.
Use small melon baller (or teaspoon) to form 1-inch round meatballs.
Roll and compress into tight balls.
In a separate wide, shallow pan, heat olive oil on medium-high heat.
Brown meatballs on all sides for about 2-3 minutes.
Depending on the pan size, you may need to brown two batches of the meatballs. Cook them in a single layer; do not crowd the pan by stacking them.
Do not overcook the meatballs; they will continue to cook in the sauce.
Towards the end of browning the meatballs, add a little bit of red wine to deglaze the pan.
Add the cheese and red wine to the sauce and stir it in. Add 1 teaspoon of salt to taste.
Add meatballs and gently stir. Simmer the sauce and meatballs for 30-45 minutes while cooking the spaghetti, stirring occasionally.
Cook the pasta and optionally add 1/3 cup of pasta water to the sauce.
Place a thin layer of sauce on a plate. Add pasta, sauce and meatballs.