Fill a medium saucepan with water and bring it to a boil. Add the chicken thighs and a bit of salt; simmer until cooked through and tender, about 20 minutes. Cool the meat; shred (discarding any fat or gristle), season with salt, and set aside.
Fill a large skillet with enough oil to submerge a tortilla (between 1/4 and 1/2 inch). Warm the oil over medium heat until a drop of water sizzles immediately. Fry each tortilla briefly in the oil, about 10 seconds per side. Use a spatula rather than tongs to flip the tortillas, as they'll tear easily. The tortilla should stay soft; if it starts to harden, it has been in the oil too long. Drain on paper towels.
Heat the oven to 400°F. With a pastry brush, spread a thin layer of sauce on both sides of each tortilla. Spoon a heaping tablespoon of shredded chicken just off center of each tortilla and roll into loose cylinders. Set the enchiladas side by side in a 9x13 inch baking dish, pour the remaining sauce over them, top with the cheeses, and bake until bubbling and browned in spots, 15 to 20 minutes.