Bake the bacon. Preheat the oven to 400°F. Line a baking sheet with parchment paper and arrange the bacon in a single layer. Bake for 18–20 minutes, or until crispy. Transfer to paper towels to drain.
Toast the brioche buns. Heat a large skillet over low heat. Place the buttered brioche buns cut-side down and toast until golden brown. Remove and set aside. Wipe the skillet clean.
Smash the burger patties. Increase the heat to medium and lightly coat the skillet with avocado oil. Working in batches, place two 4-ounce beef balls several inches apart in the skillet. Immediately press each ball firmly into an ultra-thin patty using a burger press or sturdy metal spatula. Season with salt and pepper.
Cook and melt the cheese. Let the patties cook undisturbed for 2–3 minutes, until deeply browned on the bottom. Flip, season the second side and immediately top each patty with a slice of cheddar. Reduce the heat to low and cook until the cheese melts and the beef is cooked through. Transfer to a plate and repeat with the remaining patties. Loosely tent the cooked patties with foil to keep warm.
Air-fry the french fries. While the burgers cook, prepare the frozen French fries in an air fryer according to the package directions so they're hot and crispy when the burgers are ready.
Make the smoky garlic parmesan sauce. In the same skillet, melt the butter over medium heat. Add the minced garlic and sauté until fragrant. Pour in the heavy cream and seasonings. Bring to a gentle simmer, stirring frequently, and cook for 3–5 minutes until slightly reduced and thickened. Stir in the Parmesan until melted and smooth. Add the chopped parsley, then taste and adjust with salt and pepper.
Assemble the double cheeseburgers. Add leaf lettuce to the bottom of each toasted brioche bun. Stack two cheesy smash burger patties on top, followed by two slices of crispy bacon. Add the top buns.
Dip and serve. Pour a generous pool of warm garlic Parmesan sauce onto each serving plate and add the crispy French fries alongside it. Slice each cheeseburger in half and carefully place the halves cut-side down into the sauce. Secure with toothpicks if needed and serve immediately.