Make the Vinaigrette: Combine the shallot, vinegar, oil, mustard, salt, pepper, and sugar in a medium jar with a tight-fitting lid; screw on the lid and shake vigorously to emulsify. (Alternatively, whisk the ingredients together in a bowl.)
Make the salad: In a medium nonstick skillet over medium heat, cook the bacon, stirring occasionally, until crisp, 6 to 8 minutes. Use a slotted spoon to transfer the bacon to a paper towel-lined plate to drain, leaving the fat in the pan. Season the chicken with the salt and add to the bacon fat in a single layer; cook over medium heat, stirring frequently, until cooked through, 3 to 5 minutes. Using the slotted spoon, remove the chicken to a plate and set aside to cool.
Put the romaine, chicken, and the tomatoes in a large mixing bowl. Give the dressing a shake to re-emulsify, then add three-quarters of it to the bowl; toss gently to combine. Add more dressing, little by little, until the greens are dressed to your liking. Taste and adjust seasoning, if necessary. Arrange the salad mixture on a serving platter or plates.
Sprinkle the hard-boiled eggs, avocado, blue cheese (if using), and reserved bacon over the salad. Pass the pepper mill at the table.
Make-Ahead Instructions: The vinaigrette can be made 2 days ahead and refrigerated.
Note: You can save yourself a step by purchasing packaged hard-boiled eggs at the grocery store (usually found near the fresh eggs). If you’d like to cook the eggs at home, use this proven method.