Place the two slices of sourdough in a cast iron pan so that the bottom opening of a 7.5-inch-wide chimney starter fits over the bread. If need be, trim the bread. (see note)
Lightly spritz a few wads of newspaper with vegetable oil, stuff them into the bottom of the chimney starter, and place on the grates of a grill. Fill with enough charcoal to come up to the top hole in the side of the chimney (3/4 of the way up). Light the newspaper and leave it to burn until the coals are ashy.
Combine the cheeses with the dry mustard, smoked paprika, and black pepper in a medium bowl and set aside.
When the charcoal is about half lit, place a 7 3/4-inch steel pan on top of the starter. Add 2 teaspoons of extra-virgin olive oil and the chopped onion. Season with a pinch of salt and stir occasionally until soft and showing just a bit of golden-brown.
When the charcoal is fully lit and the onions are close to done, lightly brush the bread (up-side only) with a bit more of the olive oil. Gently shake the starter to knock down any stray charcoal pieces, then place the starter over the bread in the pan for up to 45 to 60 seconds or until well browned. (Place the pan with the onion back on top of the chimney during this time.)
Carefully remove the starter, flip the bread (without brushing with oil) then replace the starter and broil until lightly charred, another 30-45 seconds. (Keep cooking those onions in the meantime.)
Remove the chimney yet again, then spoon the cooked onions evenly over the toasted bread. Top with the cheese mixture, then place the chimney starter over the bread one last time and broil another 30-60 seconds or until the cheese is melty and starting to brown. Remove the chimney, carefully close the sandwich, and remove to a plate. Rest 2 minutes before cutting in half (diagonally, of course) and serving.