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 2 tablespoons of 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 saucey, 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 3 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 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 cheese blend and the remaining 1/4 cup each of Parmesan and sour cream. Top with the remaining tortillas and 1 1/2 cups shredded 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.