Heat wine in a small saucepan over medium heat. When it begins to bubble, reduce heat to low and simmer until it is reduced to two tablespoons, about 30 minutes. Meanwhile, cover dried mushrooms with hot water and allow to soak until soft, about 10 to 15 minutes. Remove mushrooms and pour soaking liquid through a paper towel lined mesh strainer.
Place carrots in a food processor and pulse ten times. Scrape down sides. Add onion and pulse ten more times until both veggies are finely minced. Set aside in a bowl. Add softened mushrooms and pulse until finely ground, scraping down sides as necessary. Transfer to bowl. Add canned tomatoes with juice and pulse seven times to chop tomatoes. Set aside for later.
Heat oil over medium high heat in your largest skillet. Add processed vegetables and cook five minutes until soft, but not brown. Add garlic and cook 30 seconds. Add ground meat and use a wooden spatula to quickly break into large chunks. Pour the milk over the meat and begin to break the meat into half inch chunks. When milk begins to bubble, reduce heat to medium and continue breaking meat into even smaller bits. Cook about 20 minutes, until most of the liquid is evaporated.
Stir in the tomato paste. Clear a little space in the center of the pan for the sliced mushrooms and cook about one minute. Add tomatoes, mushroom soaking liquid, salt, pepper, oregano, and sugar. Simmer about 15 minutes until sauce is reduced and slightly thick.
Add reduced wine and peas. Simmer about five minutes.