Heat 1 tablespoon olive oil in a medium pot or skillet over high heat until shimmering.
Add the chicken and season with kosher salt and pepper; sear about 3 minutes per side until browned.
Add the sliced poblano pepper and cook 3–4 minutes until lightly charred.
Reduce heat to medium and add the fire-roasted tomatoes, 1/2 cup water, chopped chipotle peppers, chili powder, oregano, and cumin; stir to combine.
Simmer 15–20 minutes, until the chicken is cooked through and shreds easily, then shred the chicken in the pot and stir in the chopped cilantro; remove from heat.
Preheat the oven to 450°F (232°C). On a baking sheet, brush or rub the remaining 1 tablespoon olive oil over both sides of the tortillas.
Lay 4 tortillas flat on the baking sheet. Divide the shredded chicken evenly among them, then top each with cooked rice and shredded cheddar, and place the remaining tortillas on top.
Bake 5–8 minutes, flip each quesadilla, and bake another 5 minutes until cheese is melted and tortillas are crisp.
While quesadillas bake, stir together the Greek yogurt and lime zest; season to taste with a pinch of salt.
Serve the quesadillas topped with diced mango, pickled red onion, jalapeños, fresh cilantro, and a drizzle of the lime-yogurt sauce.
