Preheat the oven to 350°F (175°C) and line a baking sheet with parchment paper.
In a bowl, toss the tortilla strips with 1 tablespoon taco seasoning and 1 tablespoon olive oil until evenly coated.
Spread the strips in a single layer on the prepared baking sheet and bake 10–15 minutes, stirring or shaking once, until crisp and lightly browned; watch closely to avoid burning. Let cool.
Whisk together the lime juice, 1/3 cup olive oil, 2 teaspoons salsa seasoning, diced jalapeño, chopped cilantro, and salt and pepper to taste to make the dressing; set aside.
Preheat a grill or grill pan to high heat.
Place the chicken breasts in a gallon-size zipper bag or large bowl, add the packet of taco seasoning and about 3 tablespoons olive oil, and toss or shake to coat evenly.
Grill the chicken 3–5 minutes per side, or until cooked through (internal temperature 165°F/74°C), depending on thickness. Remove and rest 5 minutes, then slice.
In a large bowl, combine chopped romaine, shredded Monterey Jack, black beans, grilled corn kernels, and halved cherry tomatoes; toss to combine.
Divide the salad among 4 plates, top with sliced chicken, sliced hard-boiled eggs, avocado slices, and baked tortilla strips.
Drizzle each salad with the prepared dressing and serve immediately.