Grilled Steak and Eggs with Beer and Molasses

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For the sauce:

1 1/2 teaspoons unsalted butter

1/2 medium onion, finely diced

1 clove garlic, minced

1 cup ketchup

1/2 cup molasses

1/2 cup apple cider

2 tablespoons sugar

1/2 cup lager-style beer

1 1/2 teaspoons yellow mustard

2 teaspoons lemon juice

1 1/2 teaspoons Worcestershire sauce

1 1/2 teaspoons cayenne pepper

1 1/2 teaspoons paprika

For the steak and eggs:

2 1-pound strip steaks (1/2 inch thick)

Salt and freshly ground pepper

1 tablespoon vegetable oil

4 grilled eggs (recipe below)