You won’t be able to reproduce these exactly, not unless you’re willing to make corn tortillas, refried beans, and salsa pasilla from scratch. (The latter is a fragrant, moderately spicy salsa made from dried pasilla chiles.) You’d also need Oaxacan string cheese, which you may be able to find at a specialty market. I take a more liberal approach, calling for ingredients you might be able to find without spending a day shopping and salsa making.
Step 1: Set up your grill for direct grilling and preheat to medium. Be sure to leave a section of your grill fire-free (this is your safety zone).
Step 2: Lightly grill the tortilla to soften it, about 20 seconds.
Step 3: Brush the top of the tortilla with bacon fat. Turn it over, and using a spatula, spread a thin layer of refried beans over the entire surface. Sprinkle shredded cabbage and string cheese over half the tortilla and drizzle with salsa. Fold the tortilla in half to make a half-moon-shaped turnover.
Step 4: Brush and oil the grill grate. Arrange the quesadilla on the grate and grill until the bottom of the quesadilla is lightly browned, 20 to 30 seconds. Turn and grill the other side the same way—the filling should be hot and the cheese slightly melted. Serve at once, either whole or cut into wedges.