Add the bread flour, instant yeast and granulated sugar to the bowl of a stand mixer fitted with the dough hook. Mix briefly to combine.
Add the salt, egg and warm whole milk. Mix on low speed for 3 to 4 minutes, until the flour is absorbed and a shaggy dough forms.
Add the softened butter, about 1 tbsp at a time, allowing each addition to incorporate before adding more. Continue kneading on medium-low speed for 7 to 9 minutes, until the dough is smooth, elastic and slightly tacky. The dough should pull away from the sides of the bowl but may still stick slightly to the bottom.
Shape the dough into a ball and transfer it to a lightly greased bowl. Cover and let rise in a warm place for 45 to 60 minutes, or until approximately doubled in size.
While the dough rises, melt 1 tbsp butter in a large skillet over medium-low heat. Add the sliced onion and ⅛ tsp salt. Cook for 20 to 25 minutes, stirring occasionally, until the onion is soft and deeply golden. Add a splash of water if the onions begin sticking or browning too quickly.
Remove 1 tbsp of the caramelized onions, finely chop them and reserve them for the burger sauce. Transfer the remaining onions to a plate and let them cool completely.
Divide the ground beef into 8 equal portions, approximately 2 oz each. Gently shape each portion into a thin patty approximately 3½ inches wide. The patties should be wider than the finished filling because they will shrink as they cook.
Season both sides of the patties evenly with the Okie Rub, salt and black pepper.
Heat the same skillet over medium-high heat. Cook the patties in batches for 1 to 2 minutes per side, or until browned and cooked through to 160°F. Transfer the patties to a paper towel-lined plate and let them cool completely.
Combine the mayonnaise, ketchup, yellow mustard, reserved caramelized onion, pickle juice, Worcestershire sauce and Okie Rub in a small bowl. Refrigerate the burger sauce until ready to serve.
Line a large baking sheet with parchment paper.
Gently press the air from the risen dough and divide it into 8 equal portions, approximately 73 g each. Shape each portion into a ball. Keep the dough covered while assembling the bombs so it does not dry out.
Working with one portion at a time, roll the dough into a circle approximately 5 to 5½ inches wide. Keep the center slightly thicker than the edges to prevent the filling from breaking through the bottom.
Place half a slice of American cheese in the center of the dough. Top with one cooled burger patty, approximately 1 tbsp caramelized onions and 2 well-dried pickle chips.
Bring the edges of the dough up and over the filling. Pinch the edges firmly together to completely seal the filling inside. Place the cheeseburger bomb seam-side down on the prepared baking sheet.
Repeat with the remaining dough and filling, leaving approximately 2 inches between each bomb.
Loosely cover the assembled cheeseburger bombs and let them rise for 15 to 20 minutes. Meanwhile, preheat the oven to 375°F.
Whisk together the egg and water. Lightly brush the tops and sides of the bombs with the egg wash, then sprinkle with sesame seeds.
Bake for 15 to 18 minutes, or until the buns are deeply golden brown. The bread portion should register approximately 190°F when tested without inserting the thermometer into the filling.
Let the cheeseburger bombs cool on the baking sheet for 5 to 10 minutes. Serve warm with the burger sauce for dipping.