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. Remove the broccoli and pat dry.
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 and a few tablespoons of the dressing in a medium bowl.
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 ham, broccoli, pasta, cheese, lettuce and tomatoes. 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.)