Scoop a bit of bread out of each roll top to create a well, discarding the scooped bread. Brush the insides, top and bottom, with the melted butter. Close the rolls and wrap individually or together in aluminum foil, then place directly on the oven rack to warm while preparing the burgers, no longer than 20 minutes.
For the burgers: Add the ground chuck, bacon, paprika, salt and a few grinds of pepper to a large bowl. Gently mix with your non-dominant hand, using your fingers like a pitchfork so you don't squish the meat. Mound the mixture into a ball and divide in half, then divide each half in half again to make 4 even portions.
Add a drizzle of olive oil to a griddle pan over high heat. Gently roll each portion of meat into a ball and place on the pan. Using the back of a plate, a flat lid or a spatula (with no holes), press each ball into a flat patty, slightly larger than the rolls. Cook the patties until they easily release from the griddle, 2 to 3 minutes. Flip, then top each with a slice of cheese and continue to cook until the patties are cooked through and the cheese is melted, another 2 to 3 minutes.
Meanwhile, add the jalapeño to the griddle and char on all sides, then remove and finely chop, removing the seeds if you would like less heat.
Build the burgers: Whisk the hot sauce and maple syrup together in a small bowl. Remove the rolls from the oven and unwrap. Put a patty on each roll, then drizzle each with the maple hot sauce. Top each with some jalapeño followed by a fried egg and more hot sauce as desired.