Heat about 1/2 inch oil in a cast-iron skillet to 375°F. Working in batches and adding oil as needed to maintain level, fry cutlets, turning once, until golden brown on both sides, about 3 minutes per side. Transfer to a clean paper-towel-lined baking sheet. Season with salt and pepper when still hot.
Spread avocado on bottom bun halves. Top with fried cutlets. Mound half of the cheese on the cutlets.
Arrange onion rings on top of cheese. If using chipotles, tear into pieces and arrange on onion rings; alternatively, arrange pickled jalapeños on top of onion rings. Top with papalo and mound the remaining cheese on top. Drizzle with olive oil.
Scoop some bread from each of the top buns and discard. Close sandwiches and serve right away.
Special equipment
1 large cast iron skillet; 2 baking sheets
Notes
Cemita buns, papalo, and Queso Oaxaca can be found at well-stocked Mexican and Latin American bakeries and grocers. Queso Oaxaca should be shredded by hand into very thin strands, just like string cheese.
