Divide the ground beef into four 3 oz portions and shape each portion into a loose ball.
Place the bacon in a large skillet over medium heat and cook, turning occasionally, until crisp and browned. Transfer the bacon to a paper towel lined plate and pour off all but about 1 tablespoon of the bacon fat.
Add the onion to the bacon fat and cook for 7 to 10 minutes, stirring occasionally, until the onion becomes soft and translucent with a few lightly golden edges. The goal is grilled onions rather than deeply browned, caramelized onions. Transfer the onion to a bowl and set aside.
Smash the Burgers
Place the brioche buns cut side down in the same skillet over medium heat. Toast for 1 to 2 minutes, until golden brown with lightly crisp edges. Remove the buns and set aside.
Increase the skillet heat to medium high. Working in batches, place two beef balls in the skillet, leaving enough space between them. Cover each ball with a small piece of parchment paper and press firmly with a burger press or metal spatula to create a thin patty. Peel away the parchment paper and season the patties with the salt and pepper.
Cook the patties undisturbed for 1 ½ to 2 minutes, until a deep brown crust develops underneath and the edges become crispy. Slide a metal spatula firmly beneath each patty, scraping against the skillet to preserve the crust, then flip. Cook for 30 seconds before placing a slice of white cheddar cheese on each patty. Continue cooking for another 30 to 60 seconds, or until the cheese melts and the patties reach 165°F. Transfer the patties to a plate and repeat with the remaining beef.
Make the Garlic Parmesan Sauce
Remove the skillet from the heat and discard the excess grease. Allow the skillet to cool slightly, then wipe it clean with paper towels to remove the browned bits and dark residue. This will keep the garlic Parmesan sauce light in color rather than giving it a brown appearance.
Return the skillet to medium low and melt the butter. Add the garlic and cook for 30 to 45 seconds, stirring constantly.
Pour in the heavy cream and stir in the Dijon mustard, red pepper flakes, paprika, salt, and pepper. Bring the mixture to a simmer and cook for 3 to 4 minutes, stirring frequently, until the cream reduces slightly and begins to thicken.
Turn off the heat and stir in the Parmesan cheese until melted and incorporated. Stir in the lemon juice and fresh parsley.
Build and Dip
Assemble the burgers by layering 1 cup of arugula on each toasted bottom bun, followed by two cheesy smash patties, 4 slices of bacon and half the grilled onions. Place the top buns over the burgers.
Divide the garlic Parmesan sauce evenly between two shallow serving bowls or rimmed plates. Slice each burger in half, then place both halves cut side down into the sauce.
Arrange the fries alongside each burger. Garnish with Parmesan cheese and fresh parsley. Serve immediately so the toasted buns don't become soggy.
Notes
To bake the bacon instead, lay the slices on a rimmed baking sheet and bake at 400°F until crisp, about 15 to 20 minutes. Cook the onions in the avocado oil.
The sauce thickens as it cools. Loosen it with a splash of warm heavy cream.
USDA's safe minimum for ground beef is 160°F. At GC we cook all ground meat to 165°F.
Replace the white cheddar with white American or yellow American cheese, or use potato buns in place of brioche.