1. Preheat grill to 350° to 400° (medium-high) heat. Brush chicken with 2 Tbsp. oil; sprinkle with 1 1/4 tsp. salt and 1/4 tsp. pepper. Grill chicken, covered, 5 minutes on each side or until done. Let stand 10 minutes; chop.
2. Toss asparagus with 2 Tbsp. oil, and grill in a grill basket, covered, 1 minute on each side. Cut asparagus into 2-inch pieces. Toss together chicken, asparagus, cheese, shallot, 1/2 tsp. salt, and 1/4 tsp. pepper in a bowl. Whisk together mayonnaise, next 3 ingredients, 2 Tbsp. oil, and 1/4 tsp. salt; drizzle over chicken mixture. Top with herbs.