Prepare a medium-hot two-zone grill fire, preheat a gas grill on medium-high for 10 minutes, or heat a grill pan over medium heat.
Place chicken breasts between 2 pieces of plastic wrap and pound with a mallet or cast-iron skillet until breasts are no more than 1" thick.
Sprinkle chicken with salt and pepper. Grill, turning occasionally, until internal temperature reaches 160°F and juices run clear when cut. Transfer to a cutting board and let rest 5 minutes, then thinly slice crosswise.
Fill a large saucepan with 1” water and place a steamer basket inside. Bring water to a rolling boil, and use tongs to gently place eggs in steamer basket. Cover pan and steam eggs for 10 minutes. While eggs steam, fill a medium bowl with ice water. When eggs are done, transfer immediately to ice water to cool completely. Tap eggs all over on a countertop to crack shells, then peel and quarter lengthwise.
Heat oil in a large nonstick skillet over medium heat. Place bacon in skillet and cook, turning once, until browned and crisp. Transfer to a plate. Once cooled, roughly chop.
On a large serving platter lay out sliced chicken, eggs, bacon, romaine, avocado, tomatoes, red onion, and blue cheese.
In a blender, combine vinegar, yogurt, buttermilk, honey, and garlic. While blending, add oil in a thin, steady stream until thoroughly mixed. Add salt and pepper to taste.
Serve lettuce cup fixings with dressing for drizzling.