Prepare Smoky Aioli: Add mayo, paprika, and garlic to a small bowl and stir to combine. Refrigerate until ready to use.
Melt butter in a nonstick pan over medium heat. Whisk together eggs, hot sauce, salt, and pepper in a small bowl, until well combined.
Add eggs to pan, and stir continuously until eggs start to come together and thicken. Mix in the shredded cheese and turn heat to low. Continue stirring until the cheese has melted. Remove from heat.
Split Dinner Potato Rolls in half and lightly toast in toaster oven, if desired. Spread some aioli onto the bottom half of each roll. Evenly divide the scrambled eggs between the 4 sandwiches, topped with bacon jam, pickled jalapeños, and extra hot sauce, if desired. Serve warm.