Add the tofu to the bottom of the slow cooker. In a small bowl, combine the soy sauce, vinegar, tomato paste, garlic sauce, chili paste, Sichuan pepper, sugar and 1 cup of the water.
Heat the oil in a skillet over medium-high heat. Sauté garlic, ginger and mushrooms for about 2 minutes, or until fragrant. Add the sauce and bring to a boil. Remove from the heat and pour the mixture into the slow cooker, covering the tofu. Cook on low for 3 to 4 hours or on high for 1 to 2 hours.
About 30 minutes before serving, whisk together the cornstarch and remaining 2 tablespoons water in a small bowl and stir into the slow cooker along with the peas and carrots. Turn the heat to high and cook for an additional 30 minutes or until the sauce thickens.
Stir in the sesame oil and serve with cooked rice.