Preheat the oven to 425°F.
In a large nonstick or stainless-steel skillet over medium-high heat, add the 1 pound cubed chicken breasts or thighs and the chopped yellow onion. Cook, stirring occasionally, until the chicken is seared and the onion is softened, about 5–8 minutes.
Add the sliced poblano pepper, 3/4 cup red enchilada sauce, 2–3 finely chopped chipotle peppers in adobo, 1 teaspoon dried oregano, 1 teaspoon ground cumin, and 1 teaspoon kosher salt to the skillet. Reduce heat to medium and simmer, stirring occasionally, until the sauce thickens, about 5 minutes. Remove from heat and set the filling aside.
While the chicken simmers, make the garlic-lime sauce. Reserve 1 teaspoon of the 2 teaspoons lime zest for the mango. In a bowl, combine 1/2 cup plain Greek yogurt or sour cream, 3 tablespoons mayo, the remaining 1 teaspoon lime zest, 2 tablespoons lime juice, and the grated clove of garlic. Whisk until smooth and set aside.
Warm 4–6 flour tortillas until pliable (microwave briefly between damp paper towels or heat 15–30 seconds per side in a dry skillet).
Assemble the quesadillas: divide the 2 1/2 cups shredded Mexican cheese, the cooked chicken mixture, 1 cup cooked rice, and 1 cup chopped fresh cilantro among the warmed tortillas. On each tortilla, layer cheese first on one half, add some chicken and rice, then sprinkle more cheese and cilantro on top; fold the tortilla over to enclose the filling. Repeat until you have used the desired number of tortillas.
Place the folded quesadillas on a baking sheet and bake in the preheated oven for 8–12 minutes, turning once halfway through, until the cheese is melted and the tortillas are crispy.
Toss the diced mango with the reserved 1 teaspoon lime zest. Cut the baked quesadillas into wedges and serve topped with the garlic-lime sauce, the mango, and additional chopped cilantro.