In the bowl of a food processor fitted with the blade attachment, pulse the mushrooms until they appear coarsely ground. (This can also be done by hand.) Set aside.
Place a large heavy-bottomed pan over medium-high heat and add the oil. Add the garlic, carrots, celery and onions with a large pinch of salt and cook, stirring occasionally, until the vegetables soften and begin to caramelize, about 5 minutes.
Add the mushrooms and thyme and cook until the mushrooms begin to brown and dry out, about 5 minutes. Stir in the tomato paste and toast it until it looks rusty, a few minutes. Add the milk, season with salt and pepper and bring to a simmer. Cook, partially covered, until the sauce has thickened and all the liquid is reduced, at least 20 minutes.
Before serving, stir in some basil and freshly grated Parmesan. To serve, grate more Parmesan over top.