For the Mushroom Sauce: Adjust oven rack to center position and preheat oven to 350°F (180°C). In a large bowl, toss mushrooms with olive oil and 6 sprigs thyme, then season with kosher salt and freshly ground black pepper. Spread seasoned mushrooms on a foil-lined rimmed baking sheet. Bake until deeply browned, about 45 minutes. Discard thyme sprigs and increase oven temperature to 375°F (190°C).
In a medium saucepot, combine roasted mushrooms with 6 remaining thyme sprigs, milk, onion, and garlic and bring to a boil. Remove from heat and steep, covered, for 1 hour. Strain, pressing gently on the solids.
Wipe out pot and return to stovetop. Melt butter over medium heat until foaming subsides, then add flour and mix into a paste. Continue cooking until raw flour smell is gone, about 5 minutes. Whisking constantly, add mushroom-infused milk in a thin, steady stream, or in increments of a couple of tablespoons at a time, whisking thoroughly and getting into all corners of the pot to maintain a homogeneous texture. Sauce will initially become very thick, then get very thin once all milk is added.
Heat, stirring, until sauce comes to a simmer and begins to thicken slightly, about 10 minutes. Reduce heat to low and cook, stirring, until sauce is thick enough to coat the back of a wooden spoon, about 3 minutes. Season to taste with kosher salt and freshly ground black pepper.
For the Ground Beef: In a large skillet, heat oil over medium-high heat. Add ground beef and break up into crumbles with a whisk or potato masher. Cook until well browned, 5 to 10 minutes. Drain off any excess fat, leaving only about 1 tablespoon (15ml) in skillet along with meat.
Reduce heat to medium, add onion and garlic to ground meat, and cook for about 2 minutes. Add flour and cook an additional minute, until blond. Add chicken stock and simmer until thickened, about 3 minutes.
Remove from heat and finish ground beef mixture with peas, corn, parsley, chives, Dijon, and Worcestershire. Season to taste with kosher salt and freshly ground black pepper. Scrape meat mixture into a 2-quart baking dish.
To Finish: Top meat with mushroom sauce and arrange Tater Tots on top. Bake for 30 minutes, or until Tater Tots are crispy and golden brown. Serve with squiggles of ketchup.