Step 1In the bowl of a stand mixer fitted with the paddle attachment, combine 1 cup of the flour, the yeast and the sugar. With the mixer on low speed, add the heated milk, mixing to combine, then add the butter. Continue to mix over medium speed for 2 to 3 minutes to begin to develop the dough. Add the egg and beat for an additional minute to combine.
Step 2Switch to the dough hook, scraping down the bowl to make sure all of the ingredients are incorporated. Over low speed, gradually add the remaining flour, one spoonful at a time. Continue to mix for several minutes to develop the dough -- it will be smooth and elastic, and should clean the sides of the bowl.
Step 3Transfer the dough to a lightly oiled bowl, turning it in the bowl to oil the top of the dough. Cover and set aside in a warm place until doubled in size, about 1 hour.
Step 4Punch down the dough and turn out onto a lightly floured surface. Divide the dough into 6 even pieces (each slightly more than 4 ounces). Shape each piece into a ball, gathering the seams at the bottom of the ball and pinching them together to form a smooth round. Place the balls on a parchment-lined baking sheet, evenly spaced. Gently flatten each ball with the palm of your hand to form a disk about 4 inches in diameter. (If the dough springs back, leave it alone for a minute or so while you flatten the other balls to give it a chance to relax.)
Step 5Cover the mounds loosely with plastic wrap and set aside until puffed and nearly doubled in height, about 45 minutes. Meanwhile, heat the oven to 375 degrees.
Step 6Bake the buns until puffed and set, and golden brown on top, 15 to 18 minutes. Remove the buns from the oven and cool on the baking sheet for a couple of minutes, then transfer the buns to a cooling rack to cool completely. The buns can be stored in an airtight container for 3 to 4 days. Use a serrated knife to halve horizontally before using.
Step 1In a food processor, combine the ketchup, Worcestershire sauce, molasses, garlic, chipotle chiles, adobo sauce and blueberry jam. Puree until very smooth. This makes about 1 1/2 cups barbecue sauce, which will keep, refrigerated in an airtight container, for up to 1 week.
Step 1In a large plastic bag, combine the flour, baking powder, sea salt, pepper and Creole seasoning, shaking to thoroughly mix together. Add the onions and shake to coat. Transfer the onions to a bowl, tapping off any excess flour mixture.
Step 2In a cast iron skillet heated over medium-high heat until hot, add the bacon fat. When the fat is hot, add the onions in small batches, frying until crisp and golden brown, 3 to 5 minutes per batch. Drain the onions on a paper towel and set aside.
Step 1Heat a grill or grill pan over medium-high heat until hot. If using a grill, fit a smoker device (or foil pouch) with the mesquite wood chips to smoke while grilling.
Step 2Oil the grill top and add the burgers. Grill the burgers on each side until cooked through to desired doneness, 8 to 12 minutes. Top the burgers with cheese and continue to grill just until the cheese melts, 1 to 2 minutes.
Step 3When the burgers are almost finished grilling, toast the buns: Place the buns, cut side down, on the grill, to lightly toast. Remove the burgers and buns.
Step 4Assemble the burgers: Dress the cut sides (top and bottom) of each burger with a good spread of barbecue sauce. Top each bun bottom with a burger patty. Top the cheese with another dollop of barbecue sauce, then divide the onions, mounding a pile over each burger. Top each mound of onions with a slice of tomato, followed with 2 to 3 leaves lettuce. Top each burger with a bun top and serve immediately.