Dads Hamburger Sandwich

Ingredients

• 1 garlic clove
• 1 bay leaf
• 1½ tsp. black peppercorns
• ¾ tsp. crushed red pepper flakes
• ½ cup white balsamic vinegar or white wine vinegar
• ¼ cup sugar
• 1 medium red onion, thinly sliced
• 1 large egg yolk*
• 2 Tbsp. spicy brown mustard
• 1 Tbsp. Worcestershire sauce
• 1½ tsp. apple cider vinegar
• 1½ tsp. honey
• 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt, plus more
• 1 tsp. freshly ground black pepper, plus more
• ½ cup vegetable oil
• 2 tsp. (or more) plus 2 Tbsp. ghee or clarified butter
• 8 slices white sandwich bread
• 1 lb. ground beef
• 4 slices white cheddar
• 4 large eggs
• Tender lettuce leaves (for serving)

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