Cook the bacon in a skillet over low heat, stirring occasionally, until crisp. Remove the bacon from the pan with a slotted spoon and drain it on paper towels. Reserve the bacon fat and keep it warm.

In a small bowl, whisk together the mustard, shallot, garlic, thyme, and red wine vinegar. Whisk the extra virgin olive oil and about 2 tablespoons of the warm bacon fat into the shallot-vinegar mixture until the fats are incorporated and an emulsion forms.

Poach Eggs: Bring about 2 quarts of water to a boil in a saucepan. Add the white vinegar and lower the heat to a steady simmer. Fill a medium bowl with warm water and position it near the pot. Break each egg into a small cup or ramekin. Gently slide each egg into the saucepan of simmering water by bringing the small cup close to the water's surface and letting the egg slide out. Adjust the heat to just under a boil.Cook the eggs until the whites are cooked and the yolks are just set, about 4 minutes. Lift the eggs out of the cooking water with a slotted spoon and place them in the bowl of warm water. The eggs will stay warm in the water for a few minutes but will not overcook. (Alternatively, you can poach the eggs anyway you like and skip the warm water step.)

Place the lettuce and croutons in a large bowl and toss with enough vinaigrette to coat lightly. Reserve remaining dressing.

Divide the lettuce and croutons among two plates. Lift the eggs out of the warm water with a slotted spoon and let them drain briefly. Put two eggs on top of the greens on each plate and sprinkle each salad with the bacon pieces and parsley. Drizzle the remaining dressing over each salad and serve immediately.