Line a plate with a fresh paper towel.
In a skillet, add the bacon in one layer, turn the heat to medium, and cook, undisturbed, until the fat starts rendering from the bacon, 1 to 2 minutes. Continue to cook, stirring a few times, until bacon is crispy and brown, 4 to 5 minutes. Remove bacon to drain on the prepared plate. Pour off (and save for another use) all but 2 tablespoons of rendered bacon fat. Set aside the skillet with the remaining bacon fat.
Fill a saucepan with about 2 inches of water. Bring to a boil. Reduce the heat to a gentle simmer. Working one at a time, gently crack an egg into a small bowl or cup. Place the bowl close to the surface of the hot water and gently tip the egg into the water. Swirl the water gently with a spoon for a few seconds, just until the egg begins to set. Repeat with remaining egg. Cook, gently swirling the water occasionally, until the egg whites are fully set but yolks are still soft, 3 to 4 minutes. Carefully lift the eggs from the water with a slotted spoon, blot the bottom of spoon on a tea towel, then transfer the egg to a shallow bowl; reserve.
While the eggs are cooking, make the dressing: heat the pan with the bacon drippings over medium heat. Add the vinegar and mustard, whisk, and season to taste with salt and pepper.
Place the frisée in a large bowl and dress with two-thirds of the dressing. Divide dressed frisée between salad or dinner plates. Sprinkle the bacon over, then top each salad with a poached egg. Spoon the remaining dressing over each egg and salad. Season with salt and pepper, garnish with the chives, and serve.