For the dough: Brush a medium bowl with bacon fat and set aside.
In a large bowl, whisk together the water, sugar and yeast until dissolved. Add the bacon fat and egg and whisk to combine. Add the salt and 2 cups of the flour and stir to combine. Add 1 more cup of the flour and stir until the dough comes together in a rough ball, adding the remaining flour 1 tablespoon at a time as needed. Turn the dough out onto a clean work surface and knead briefly; the dough does not have to be totally smooth.
Place the dough in the prepared bowl and turn to lightly coat it in the bacon fat. Cover with plastic wrap and set in a warm place until the dough doubles in size, about 1 hour. Gently deflate the dough and turn out onto a clean work surface. Portion into 12 equal pieces.
For burger buns, line 2 baking sheets with parchment paper. Using a cupped hand, roll each portion into a smooth ball then transfer to the prepared baking sheets and gently press down on each to form a 2 1/2-inch-wide disk.
For hot dog buns, line 1 baking sheet with parchment paper. Roll each portion into a tight 5-inch-long log. Arrange the logs lengthwise in 2 rows of 6, parallel to the short ends of the sheet, leaving a few inches between the short sides of the sheet and between the rows.
For either bun, cover the baking sheet(s) with plastic wrap sprayed with cooking spray and let rise until doubled in size, about 30 minutes.
For finishing: Preheat the oven to 375 degrees F.
In a small bowl, whisk together the egg, pepper and mustard powder. Brush onto the tops of the buns. Sprinkle with the Cheddar and bacon.
Bake until golden brown, about 20 minutes. Let the buns cool on the baking sheets.