Heat a medium skillet over medium-high heat. Use a spatula to crumble the chorizo as you place it in the pan. Sauté the chorizo until browned and cooked through, about 5 minutes. Using a slotted spoon, remove the chorizo from the skillet to a plate.
Arrange 5 tortillas on a work surface. Spread about 1/4 cup refried beans evenly over half of each tortilla, leaving a 1/2-inch border. Scatter about
1/3 cup cooked chorizo over beans. Scatter a generous 1/3 cup of cheese over top. Divide and arrange cilantro over top of cheese. Fold opposite half of tortilla over the filling. (The quesadillas can be assembled up to 1 hour before serving. Cover loosely with plastic wrap so they don’t dry out, and set aside at room temperature.)
Prepare a medium fire in a charcoal grill or preheat a gas grill on medium. Brush outsides of tortillas with a small amount of oil. Using a wide spatula, transfer quesadillas to the grill. Grill on one side for about one minute, slide and turn quesadillas 90 degrees, grilling until nice cross-hatch grill marks appear. Slide spatula underneath to flip quesadillas and grill the second side. Transfer the quesadillas to a cutting board. Use a sharp knife or pizza cutter to cut the quesadillas into 5 wedges. Arrange on a platter and serve immediately accompanied with the tomatillo salsa.