Heat the olive oil in a large skillet over medium high. Once hot and shimmering, add the turkey, onion, bell pepper, salt, and pepper. Cook, breaking up the meat, until the turkey is cooked through and most of the excess liquid has evaporated, about 8 minutes. Add the chili powder, cumin, and cayenne and cook until fragrant, about 30 seconds.
Add the black beans, pinto beans, diced tomatoes in their juices, and carrots. Cook, stirring occasionally, until the tomatoes are broken down and most of their liquid has cooked down, about 5 minutes. Remove from heat. Taste and adjust seasoning as desired.
Lightly coat a 5-quart or larger slow cooker with nonstick spray. Spread 1 1/2 cups of the turkey-bean mixture in an even layer on the bottom. Top with 2 tortillas, spreading the tortillas out so that they cover the beans, overlapping them in the middle as needed. Sprinkle with 1/2 cup shredded cheese. Continue layering as follows: 1 1/2 cups beans, 1/2 cup cheese, 2 tortillas, 1 1/2 cups beans, 2 tortillas, 1/2 cup cheese, remaining beans. Reserve the final 1/2 cup cheese to sprinkle on top.
Cover and cook on low for 4 to 5 hours or on high for 2 to 3 hours, until heated through. Sprinkle the remaining 1/2 cup cheese on top. Cover, turn the heat to high, and cook until the cheese melts, 5 to 10 additional minutes. Cut into slices and serve warm with desired toppings.