Whisk 0.25 cup balsamic vinegar, 2 tablespoons olive oil, honey, and oregano in a bowl; marinate chicken for at least 30 minutes.
Grill chicken over medium-high heat for 6 to 7 minutes per side until the internal temperature reaches 165°F (74°C).
Rest chicken for 5 minutes before slicing into 0.5-inch strips.
In a small bowl, emulsify the remaining balsamic vinegar, extra-virgin olive oil, Dijon mustard, salt, and pepper to create the dressing.
Distribute the chopped romaine lettuce evenly across four bowls or a large platter.
Arrange the sliced chicken, eggs, avocado, bacon, tomatoes, and red onion in distinct rows over the bed of lettuce.
Garnish with blue cheese crumbles.
Serve immediately with the balsamic vinaigrette on the side or drizzled over the top.