Make the Blue Cheese Grits: In a medium saucepan with a lid, bring the stock, milk, garlic, and pepper up to a boil.
Slowly whisk in the grits. Turn the heat down to low, just barely a simmer, and place the lid on the pot. Cook the grits for 15 to 20 minutes, or until the liquid has been absorbed.
Turn the heat off and add the cheese. Stir until the cheese is melted. Taste for seasoning and add salt if needed. If your stock and cheese have ample salt, you may not need to add any additional salt.
Keep warm.
Make the Mushrooms: In a shallow skillet, heat 1/2 tablespoon of the oil and 1/2 tablespoon of the butter. Heat over medium-high heat until the butter is melted.
Add half of the mushrooms, a pinch of salt and pepper and cook, stirring occasionally, until the mushrooms are soft and browned. Remove them to a plate.
Repeat with the other 1/2 tablespoon of oil and 1/2 tablespoon of butter. Cook the remaining mushrooms, then add them to the first batch of cooked mushrooms.
Lower the heat to medium, and add 1 tablespoon of butter to the pan. Once melted, add 1 tablespoon of flour. Whisk until no lumps of flour remain, and the butter is bubbling.
Slowly add the beef stock, whisking to combine well. Add the thyme and stir again.
Let the mixture come up to a boil, raising the heat slowly if necessary.
Once the mixture is boiling, it will thicken slightly. Add the mushrooms back to the pan to heat through. Taste for seasoning and adjust as needed. Remove the thyme stems and discard.
Serve the mushrooms over the Blue Cheese Grits. Enjoy!!