Slice the fresh button mushrooms into thin slices.

Add the oil to a skillet and heat it over medium heat. When the oil gets hot (after a few moments), add the ground beef and saute it with a wooden spoon, stirring frequently to break up the big chunks of meat into smaller ones.

Once the beef is no longer pink, add the sliced mushrooms. Continue sauteing, stirring often, until the water has been released and the mushrooms become pale brown. This step may take several minutes, depending on the thickness of the mushrooms.

Sprinkle all-purpose flour and continue sauteing them for about one more minute, stirring frequently. Make sure the ground beef and mushrooms are well-coated with the flour.

Add the dried thyme, parsley flakes, salt, and pepper. Stir to combine. Then pour in the milk, stir again, and let it simmer for 2 to 3 minutes. Add the allium-free mustard and Greek yogurt.

Simmer the sauce, stirring occasionally, until it thickens. Taste and adjust the salt and pepper, if needed.