Put the oil in a skillet over medium heat. When the oil is hot, add the bacon and cook slowly until it’s crisp all over, 10 minutes or more. Add the shallot and cook until softened, another minute or 2. Keep the bacon warm in the skillet.
Warm a salad bowl by filling it with hot water and letting it sit for a minute. Dry it and toss in the greens. Add the vinegar and mustard to the bacon and bring just to a boil, stirring. Pour the liquid and bacon over the greens, toss, and season to taste; it shouldn’t need much salt. Top each portion with an egg and serve immediately.