In a small bowl stir together onion, vinegar, mustard, and a pinch salt; let stand at least 30 minutes.
Meanwhile, season chicken on both sides with salt and black pepper; brush with 1 Tbsp. olive oil. Grill on rack of a covered grill directly over medium-high heat 8 to 10 minutes or until 165 degrees F, turning once.
Divide chicken, egg halves, avocado, and blue cheese over romaine among four plates. Whisk remaining 2 Tbsp. olive oil into onion mixture; drizzle over salads. Sprinkle with bacon.