Preheat a grill to medium high. Cook the bacon in a large nonstick skillet over medium-high heat, turning, until crisp. Drain the bacon on paper towels, reserving 2 tablespoons bacon drippings.
Form the beef into four 6-ounce patties, about 3/4 inch thick; press the center of each patty with your thumb to make a 1/2-inch-deep indentation. Season the patties on both sides with salt and pepper. Lightly brush the grill grates with vegetable oil, then grill the patties, indentation-side up, until marked on the bottom, about 5 minutes. Flip and cook until marked and slightly firm, about 3 more minutes for medium doneness. Remove the patties to a plate and let rest 5 minutes. Grill the buns, cut-side down, until lightly toasted, about 1 minute.
Heat 2 tablespoons olive oil in the same skillet over medium-high heat. Add the eggs and fry until the yolks are set but still runny, about 5 minutes. Season with salt and pepper.
Whisk the mustard, the remaining 1 tablespoon olive oil, the reserved 2 tablespoons bacon drippings and the shallot in a medium bowl. Season with salt and pepper. Add the frisee and toss; season with salt and pepper.
Spread the bun bottoms with mustard. Top with the burger patties, bacon, frisee salad and fried eggs. Cover with the bun tops.