Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat. Brown the ground turkey and beef (or chicken), breaking it up into small crumbles. Drain the excess fat. Add marinara sauce, smoked paprika, oregano, red pepper flakes (if using), salt, and pepper. Simmer for 5 minutes.
In a medium bowl, combine ricotta cheese, goat cheese, half of the shredded mozzarella, and half of the Parmesan. Add fresh basil, salt, and pepper, stirring until fully incorporated.
In a prepared baking dish, spread a thin layer of the meat sauce. Place a tortilla on top, then add a layer of cheese mixture. Continue layering: sauce, tortilla, cheese, until all ingredients are used, finishing with a layer of sauce.
Top the final layer with the remaining mozzarella, Parmesan, and fresh basil leaves.
Cover the dish with aluminum foil and bake for 20 minutes. Remove foil and bake for an additional 15-20 minutes until cheese is golden brown and bubbly.
Let the lasagna rest for few minutes before slicing and serving.