Heat 1 tablespoon of the olive oil in a medium pot or large 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 and cook 3–4 minutes until slightly charred.
Reduce heat to medium and add the fire-roasted diced tomatoes (with juices), 1/2 cup water, chopped chipotles, chili powder, oregano, and cumin; stir to combine.
Simmer the mixture 15–20 minutes, until the chicken is cooked through and easily shreds.
Shred the chicken directly in the pot, remove from heat, and stir in the chopped cilantro.
Preheat the oven to 450°F (232°C). On a baking sheet, brush or rub the remaining 1 tablespoon olive oil over the tortillas.
Place four tortillas on the sheet, then evenly layer each with a portion of shredded cheese, cooked rice, and the shredded chicken mixture; top each with the remaining tortilla.
Bake the quesadillas 5–8 minutes, then flip and bake an additional about 5 minutes, until cheese is melted and tortillas are crisp.
Meanwhile, stir together the Greek yogurt and lime zest and season with salt to taste.
Slice or quarter the quesadillas and serve topped with diced mango, pickled red onion, jalapeños, fresh cilantro, and a drizzle of the lime yogurt.