Bring a large pot of salted water to a boil. Add the pasta and cook until al dente (about 2 minutes less than the label directs), adding the broccoli during the last 4 minutes of cooking. Drain the pasta and broccoli and rinse under cool water; shake off the excess. .
Whisk the mayonnaise, buttermilk, 1/4 cup chives, the parsley, half of the lime juice, 1/4 teaspoon salt, and pepper to taste. Toss the pasta with some of the dressing in a medium bowl. Reserve the remainder of the dressing to drizzle over the top of the salad.
Assemble the salad: Toss the avocados with the remaining lime juice in a large glass serving bowl and season with salt; arrange in an even layer. Top with layers of the chicken, broccoli, pasta, cheeses, lettuce, tomatoes, corn and poblano pepper slices. Drizzle some of the remaining dressing on top and sprinkle with the remaining 1 tablespoon chives, or cover and refrigerate the salad and dressing separately up to 6 hours. (Bring to room temperature before serving.)