In a large bowl, combine the beef, 2 of the egg yolks, buttermilk, Worcestershire sauce, Baby Bam, garlic, Italian Essence, and salt. Using a wooden spoon or your clean hands, stir well to combine.
Divide the meat into 4 equal portions, about 3/4 cup each. Pat into patties 3 1/2 inches in diameter and 1 inch thick each. Wash your hands.
Heat a large skillet over medium heat and, when hot, cook 2 patties at a time until browned on both sides but not cooked all the way through: 2 minutes on the first side, and 1 minute on the second side.
Remove the patties from the skillet and set aside on a plate to cool. Repeat with the remaining patties. Refrigerate the patties for 30 minutes.
Position rack in center of oven and preheat the oven to 425Â°F. Line a large baking sheet with parchment paper and set aside.
While the meat is cooling, lightly dust a work surface with the flour. Roll out one puff pastry sheet to about 1/16-inch thickness and cut four 6-inch rounds from the sheet.
Repeat with the remaining puff pastry sheet.
In a medium bowl, combine the remaining egg yolk with the water and beat with a whisk or fork to make an egg wash.
Place 4 of the pastry rounds on the prepared baking sheet and, using the pastry brush, paint a 1/2-inch border of the egg wash around each of the 4 rounds.
Place a chilled hamburger patty in the middle of the round and top each with 1 slice of cheese, letting the edges hang over the sides of the patty.
Top the cheese with the remaining pastry rounds, pressing the edges together with a fork to seal.
Paint the top and sides of each pastry round with the egg wash and, using the tip of a paring knife, cut a small "x" in the top of each Wellington to allow steam to escape.
Bake the Wellingtons until golden brown, about 15 to 20 minutes.
Using oven mitts or pot holders, remove the baking sheet from the oven and serve the Wellingtons immediately.
YIELD: 4 servings