Preheat the oven to 375 degrees F. Line a baking sheet with foil. Toss the tortilla strips with 1 tablespoon vegetable oil, 1/2 teaspoon lime zest, a pinch of salt and a few grinds of pepper in a medium bowl; spread in a single layer on the prepared pan. Bake, stirring halfway through, until browned and crisp, 10 to 12 minutes; let cool.
Meanwhile, toss the beans, corn, tomatoes, pickled jalapenos and brine, the remaining 1 tablespoon vegetable oil and 3/4 teaspoon lime zest, the lime juice, 3/4 teaspoon salt and a few grinds of pepper in a medium bowl. Let sit, tossing occasionally, until juicy, about 10 minutes.
Divide the lettuce among 4 bowls and top with the bean mixture and juices, chicken, avocado, cheese and tortilla strips.