Bring a large pot of water to a boil, add in about 1/4 c of salt, then add in the pasta. When the water comes back up to a boil, begin the timer for about 7 minutes—this will cook the pasta al dente. Remove from heat, drain noodles, then place back in pot with a cover to keep warm—do not add oil.
Coat the bottom of a large skillet with olive oil. Heat oil on medium, then, when oil is hot, add in the mushrooms. Allow mushrooms to brown for about 4-5 minutes, then add in the grated carrot, garlic, oregano, tarragon, beef, chili, salt, and pepper. Brown the beef for about 5 minutes.
Next, add in tomato paste, and sugar. Stir until paste is evenly dispersed, then add in the pasta, wine, vinegar, and lemon juice. Allow to simmer for 10 minutes.
Mix beef and pasta together and serve warm.
Garnish with kalamata olives, cherry tomatoes, basil, thyme, and lemon zest. Serve with fresh Parmesan cheese!