Place a large heavy-bottomed pan over medium-high heat and add the oil. Add the meat and a large pinch of salt and cook, breaking up any lumps, until it begins to brown, about 5 minutes.
Add the garlic, celery, carrots, onions and thyme and cook, stirring occasionally, until the vegetables soften and begin to caramelize, about 5 minutes.
Stir in the tomato paste and toast it until it looks rusty, a few minutes. Deglaze the pan with the white wine, scraping up any brown bits from the bottom of the pan with a wooden spoon. Add the milk and bring to a simmer. Cook, partially covered, until the sauce has thickened and all the liquid is reduced, at least 20 minutes. Season with salt and pepper and remove the thyme bundle.
Before serving, stir in some freshly grated Parmesan. To serve, grate more Parmesan over top.