Melt unsalted butter (margarine or a 53% vegetable oil spread) in a large saucepan over medium heat. Add chopped onions, thinly sliced mushrooms, salt, coarse ground black pepper, and dried thyme. Stir until well combined.
Reduce heat to low. Continue stirring until onions and mushrooms are soft.
Lightly sprinkle all-purpose flour all over top of mixture. Pour in HOT turkey (or chicken) broth. Stir constantly to prevent clumping.
Over low heat, simmer 12 minutes. Mixture will be slightly thick.
Slowly pour in whole milk (or heavy whipping cream). Stir until well combined.
Over low heat, Simmer 3 more minutes OR until bubbles appear around edges.
Remove from heat immediately.
Soup will thicken as it cools.
Serve warm or hot.