Step 1

For the puffs: Preheat the oven to 400 degrees F. Line an 18-by-13-inch rimmed baking sheet with parchment paper.

Step 2

In a medium saucepan, bring the butter, salt and 1 cup water to a boil over medium-high heat. Decrease the heat to low. Add the flour and stir constantly until a sticky ball of dough forms, about 4 minutes.

Step 3

Transfer the mixture to the bowl of a stand mixer fitted with the paddle attachment and mix on medium-low speed until there is no longer any steam. Decrease the mixer speed to low and add 3 of the eggs, one at a time, mixing after each addition. The dough will be done once it clings to the paddle in "spokes" as it spins in the bowl. You may need to add another egg to get this consistency. Add the Gruyere and mix to combine.

Step 4

Dab a small amount of the dough under the parchment paper at the four corners and in the center of the prepared baking sheet to keep the parchment paper in place. Using a 3-ounce ice cream scoop, scoop out the dough onto the parchment, 2 inches apart. You will end up with 12 scoops. In a small bowl, whisk the remaining egg with a splash of water and a pinch of salt. Brush each cream puff with the egg wash and sprinkle with the sesame seeds. Bake for 20 minutes, then rotate the baking sheet 180 degrees. Lower the oven temperature to 375 degrees F and bake until deep golden brown, about 20 minutes. Transfer the cream puffs to a wire rack and allow to cool to room temperature.

Step 5

For the cheese mousse: Meanwhile, add the cream cheese to a bowl, and using a hand mixer, beat until soft, about 1 minute. Add the processed cheese, paprika, salt and white pepper and beat until fluffy and smooth. Mix in the heavy cream as needed until the mixture is velvety and smooth but not too loose. Transfer to a piping bag with a large star tip. Set aside.

Step 6

For the burger patties: Transfer the ground beef to a cutting board. Cut into 12 squares, then roll each square into a ball. Place onto a piece of wax paper. Fold the wax paper over, leaving room for the meat to spread out. Using a rolling pin, roll the patties until paper thin. Season with salt and black pepper.

Step 7

Heat a large nonstick skillet over medium-high heat. Add the butter and, once melted, cook the patties for 4 minutes on the first side. Flip and continue to cook for another minute or two. Remove to a plate. Add the onions to the same skillet and cook over medium heat, stirring occasionally, until golden brown and translucent, 15 to 20 minutes. Remove from the heat.

Step 8

To assemble: In a small bowl, mix the ketchup, mayo and sweet relish together. Set aside. Slice the puffs in half. Spread the special sauce on the bottom half of each puff, then top with a patty, onions, bacon if using and lettuce. Before topping each burger with the top "bun," pipe some of the cheese mousse into it first. Top each burger cream puff with a halved cornichon secured with a toothpick.