Cook the nopales: In a medium pot, add the diced cactus paddles, 2 cloves garlic, 1 tablespoon kosher salt, ¼ teaspoon Mexican oregano, and enough water to completely cover them. Bring to a boil over high heat, reduce heat to low, cover, and simmer for 10 minutes.
Drain the water from the nopales and rinse them under warm water to remove any excess slimy liquid.
While the nopales are cooking, make the chile rojo: In a medium pot, add the guajillo chiles, árbol chiles, and enough water to completely cover them. Bring to a boil over high heat. Remove the pot from the heat, cover, and let the chiles soak for 10 minutes to soften.
Using a slotted spoon, transfer the softened chiles to a blender. Add the tomato, the remaining 1 clove garlic, the remaining 1 teaspoon kosher salt, the remaining ¼ teaspoon Mexican oregano, and ½ cup fresh water. Blend until completely smooth. If the sauce is not smooth, strain it through a fine mesh strainer to remove any solids.
Heat the oil in a large saucepan or skillet over medium-high heat. Add in the sliced onion and cook for 2 minutes, stirring occasionally.
Add in the drained nopales and cook for 5 minutes, stirring occasionally.
Add the chile rojo and cook for 6 more minutes until the sauce has thickened. Taste and season with more salt as needed. Remove from heat.
Assemble and cook: Layer the half of each tortilla with ⅓ cup cheese, ⅓ cup cooked nopales, and another ⅓ cup cheese in each tortilla. Fold it over and cook the quesadillas in a hot skillet or comal for 1 to 2 minutes per side until the cheese has fully melted and the quesadillas are golden brown.
Make sure to cook down the nopales en chile rojo until the sauce has thickened. This will ensure that the filling doesn’t completely run out of the quesadillas during cooking.
Don’t take the quesadillas off the skillet too soon. The quesadillas should be golden brown and a little crispy on the outside.