Cookwith - Absurdist Filet Mignon Breakfast Burger with Tangy Shallot Jam

Ingredients

• 4 filet mignon steaks (about 4 oz each, 1-inch thick)
• 4 soft brioche buns
• 4 large eggs
• 2 tbsp olive oil
• 2 tbsp unsalted butter
• 1 large shallot, thinly sliced
• 1/4 cup red wine vinegar
• 1 tbsp sugar
• 1 cup baby arugula
• 1/4 cup mayonnaise
• 1 tbsp Dijon mustard
• 1 tsp lemon juice
• Kosher salt, to taste
• Freshly ground black pepper, to taste