Hard-cook eggs, then cool, peel, and quarter. Meanwhile, whisk together 1/3 cup oil, the vinegar, mustard, 3/4 tsp. salt, 1/2 tsp. pepper, the garlic, chives, and parsley; set dressing aside.
Put asparagus at one end of a large rimmed baking sheet and green onions at other end. Toss asparagus with 1 tbsp. oil and 1/4 tsp. each salt and pepper. Toss onions with remaining 1/2 tbsp. oil and a sprinkle of salt and pepper.
Trim bases from lettuces and halve or quarter lettuces if large. Arrange in a bed on a large platter and set aside.
Heat a grill to medium-high (about 450°). Lay cold prosciutto slices on two sheets of parchment paper, separated; then stack on a rimmed baking sheet.
Carefully flip prosciutto sheets meat down onto grill. Pull off paper with tongs. (For a charcoal grill, cook one batch at a time and move from center of fire if meat starts browning too fast.) Grill prosciutto covered, turning once, until mostly crisp, about 2 minutes. Transfer to baking sheet; set aside.
Grill asparagus and green onions, covered, turning once, until charred and tender, 4 to 6 minutes. Transfer to baking sheet.
Arrange asparagus, green onions, prosciutto, avocados, eggs, and cheese over lettuce. Drizzle with some of reserved dressing and serve the rest on the side.