Preheat the oven to 425 degrees F. Put the potatoes in a saucepan, cover with cold water and season with salt. Bring to a boil and cook until the potatoes are tender, 6 to 8 minutes, adding the broccoli halfway through. Drain and transfer the potatoes and broccoli to a large bowl; set aside.
Meanwhile, set a wire rack on a rimmed baking sheet. Pulse the corn chips in a food processor until finely ground. Transfer the crumbs to a shallow bowl. Season the chicken on both sides with salt and pepper. Spread the chicken with 1/4 cup sour cream, coating both sides, then press into the corn chip crumbs to coat; transfer to the rack. Bake until crisp and cooked through, about 18 minutes.
Add the tomatoes and olive oil to the potatoes and broccoli; season with salt and pepper and toss. Mix the remaining 1/2 cup sour cream, the scallion, cilantro, lime juice, 1/2 teaspoon salt and 2 tablespoons water in a small bowl. Drizzle over the vegetables and serve with the chicken.
You can use this recipe to make chicken fingers, too. Cut the breasts diagonally into strips before coating. Bake as directed.