Preheat the oven to 350 degrees F. Brush the inside of an 8-inch square baking dish with oil.
Toss the chicken with 1/2 teaspoon salt in a medium bowl. Heat the oil in a large nonstick skillet over medium-high heat. Add the chicken and cook, undisturbed, until it begins to brown in spots, about 2 minutes. Add the poblanos, tomatoes and 1/2 teaspoon salt and cook, stirring frequently, until the chicken is cooked through, the peppers are crisp-tender and the tomatoes are saucy, 5 to 6 minutes. Remove the skillet from the heat and stir in the garlic. Transfer the chicken, vegetables and juices to a medium bowl to cool.
Wipe out the skillet, add enough oil to just cover the bottom and heat over medium heat. Once the oil begins to shimmer, add 3 of the tortillas and fry until they are hot but still pale, about 10 seconds per side. Remove the tortillas to a baking sheet and sprinkle with salt. Fry the remaining tortillas in batches of 3, adding and heating oil as needed between batches.
Cover the bottom of the prepared baking dish with 4 overlapping tortillas. Top with half the chicken-poblano mixture, sprinkle with 1 cup of the Mexican cheese blend and 1/4 cup of the Parmesan and dollop with 1/4 cup of the sour cream. Top with 4 more overlapping tortillas, the remaining chicken-poblano mixture, 1 cup of the remaining Mexican cheese blend and the remaining 1/4 cup each of Parmesan and sour cream. Top with the remaining tortillas and 1 1/2 cups Mexican cheese blend.
Cover the dish loosely with foil and bake until the lasagna is hot and the cheese is melted, about 25 minutes. Remove the foil and bake for 5 minutes more. Let rest for a few minutes. Cut into 4 squares, place on 4 serving plates and divide the romaine and radishes among the plates.
Copyright 2016 Television Food Network, G.P. All rights reserved.