Melt butter in large skillet. Add thinly sliced onions, thyme and pepper and cook covered for 13-15 minutes, stirring occasionally.
Uncover and cook and stir over medium high heat for 3 - 5 minutes more or until onions are golden. Stir in 1/2 cup Shredded Smoked Gouda or Cheddar to melt.
Divide ground beef into 4 patties and cook to 160. Serve burgers with cheesy onion mixture on a bun
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