Bring a large pot of water to a boil.

Heat oil in a large skillet over medium until shimmering, add bacon and cook, stirring frequently,  until crisp, about 5 minutes.

Meanwhile, crack eggs into a fine-mesh sieve, let drain 30 seconds, then gently transfer to a liquid measuring cup. 

Stir vinegar and 1 teaspoon salt into boiling water. Remove pot from heat, then gently lower liquid measuring cup to just above  water and carefully drop eggs into water one at a time, rotating 90-degrees after each addition. Cover pot and let eggs cook 3 minutes until whites are set. If whites are not set after 3 minutes, continue to cook, checking every 30 seconds. 

Gently remove each egg with a slotted spoon, allowing water to drain off, transfer to a plate and set aside.

Drain all but 2 tablespoons bacon grease from skillet. Add shallot to skillet with crisped bacon; sauté over medium heat until softened, about 2 minutes. Stir in vinegar and mustard; scraping up any brown bits. 

Remove skillet from heat, add greens and toss to evenly coat. Season salad with salt and a generous amount of cracked pepper. Toss with any reserved bacon grease to taste, optional. Divide salad between four bowl and top each with a poached egg or two.