In a small saute pan, cook the bacon over medium-low heat until almost crisp and the fat has rendered. Remove the bacon to a paper towel-lined plate, leaving the rendered fat in the pan. To the pan, whisk in the mustard, white wine vinegar, olive oil, salt, and black pepper, to taste. Remove the pan from the heat.
Fill a medium saucepan 2/3 with water. Add the distilled white vinegar and bring to just barely a simmer. Cook's Note: like very active club soda.
Crack an egg and place it into a small ramekin. Very slowly and gently place the egg into the water by placing the ramekin right next to the water and letting the egg slide out of the dish and directly into the water. The ramekin should be touching the water. Repeat with the remaining eggs. Each egg should cook for 3 1/2 minutes, exactly.
In a large bowl, toss the greens with the tomatoes in the bacon dressing and divide the mixture among 4 salad plates. When the eggs are ready, carefully remove with a slotted utensil, and blot briefly on a paper towel to remove the excess vinegary water. Gently place an egg on top of each salad. Grind some black pepper on top of the eggs, top with lardons and serve.