Preheat the oven to 425°F and line a rimmed baking sheet with parchment paper.
Heat 2 tablespoons olive oil in a 12-inch skillet over medium heat.
Add the ground chicken, diced onion, diced red bell pepper, chili powder, smoked paprika, cumin, kosher salt, and black pepper. Cook, breaking up the meat, until the chicken is browned and the vegetables are softened, about 6–8 minutes.
Add the minced garlic and cook for 1 minute more.
Reduce heat to low and stir in the drained black beans, chopped green chilies, chopped spinach, frozen corn, chopped cilantro, lime juice, and lime zest until combined.
Gently simmer on low, stirring occasionally, until the spinach is wilted and most moisture has evaporated, about 8–10 minutes.
Remove the skillet from the heat and fold in the shredded pepper jack cheese until evenly distributed.
Lay out the 10 tortillas and divide the filling evenly, placing each portion down the center of a tortilla and leaving about 1 inch clear at each end.
Fold in the short ends of each tortilla and roll tightly to enclose the filling; place each seam-side down on the prepared baking sheet.
Brush the tops of the rolled tortillas with the remaining 2 tablespoons olive oil.
Bake for about 15 minutes, or until the tortillas are golden brown and crisp.