Stack tortillas, cut in half, and cut crosswise into ½" strips. Place tortilla strips on a rimmed baking sheet.
Toss with cumin, 1 Tablespoon of the olive oil, and ¼ teaspoon of the salt.
Spread strips in a single layer, bake in a preheated oven until golden brown and crisp, 10-12 minutes, turning halfway through baking. Cool strips to room temperature.
Meanwhile whisk together lime zest, lime juice, shallot, apple cider vinegar, Dijon mustard, honey, pepper, and remaining ⅓ cup oil and ¾ teaspoon salt until well combined. Pour through a fine strainer.
Toss together chopped chicken, avocados, and romaine. Add vinaigrette to taste. Top with tortilla strips.