Add carrots, celery, shallots and garlic to a food processor and pulse until finely broken down. Be careful not to break the ingredients down too much, or it'll become a pulp!
In a pot over medium heat, add oil then the food processor contents. Season with salt and sauté for 2-3 minutes, or until nice and aromatic. Add the vegan mince (see notes if using tofu, or substitute) and sauté for another 2-3 minutes, until a slight browning occurs.
Add the white wine to the pot, and continue to cook until the wine has evaporated, about 5-7 minutes. Add the tomato paste, cooking down until it has darkened in colour, about 3-4 minutes.
Add bay leaf, thyme, tamari, vegetable broth, milk and black pepper, stir to combine and drop the heat to a low simmer. Let the mixture cook, stirring occasionally, for about 18-25 minutes, or until it has reached a sloppy joe texture and has thickened up considerably.
To serve, cook pasta of choice according to box directions. When ready to serve, toss in sauce along with a splash of pasta cooking water and stir to coat. Top with a generous amount of vegan parmesan and enjoy.