For the twists: Preheat the oven to 375 degrees F. Line a baking sheet with parchment paper and spray lightly with nonstick spray.
Whisk together the eggs, mustard and a splash of water until smooth; sprinkle with salt and pepper. Set aside.
Dust a work surface with flour and roll out the puff pastry to a 10-by-14-inch rectangle with the shorter side facing you. Spread the cheese over the top half of the dough, then fold the bottom half over the cheese half and roll lightly to seal. Cut the dough vertically into 12 even strips; each strip will be slightly thinner than 1 inch.
Place a bacon strip on each pastry strip and twist the pastry and bacon together. Place the twists on the prepared baking sheet, pressing down the ends. Brush the tops with some of the egg wash and sprinkle with salt and sesame seeds or poppy seeds (or both).
Bake until the pastry is golden brown and the bacon is crisp, 12 to 15 minutes. Remove the baking sheet to a rack and let cool for 5 minutes before removing.
For the eggs: Bring a small saucepan of water to a boil over medium-high heat; there should be enough water to cover the eggs by at least 1 inch. Using tongs, gently place eggs in the water and boil for 5 1/2 minutes for soft-cooked eggs; the whites will be set, but the yolks will be very runny.
Transfer the saucepan to the sink, pour off most of the water, and then run under cold water for 30 seconds. Set the soft-boiled eggs in egg cups; use an egg topper to slice off the tops of the shells, or tap around the shells with a small spoon to break and peel off. Season lightly with salt and pepper, and serve with the twists for dipping in the yolks.