In a large pot, melt the butter until it is frothy and browned. Add the sausages and brown on all sides, about 1-2 minutes per side. It's okay if the brats are not cooked through. Leave the brats in the pot.
Add the sliced onion to the pot and stir. Cook until tender, stirring occasionally, until the onions are tender and browned - about 10 minutes.
Pour the beer into the pot and bring to a rolling simmer. Boil the brats for 15-20 minutes (depending on the type of brat you are using) until cooked through and then remove the brats from the pot and set aside to dry.
Continue to simmer the onions and beer until the liquid has evaporated entirely, leaving a dark, sweet and sticky sauce - about 30-40 minutes.
Meanwhile, stir together the ketchup, mayonnaise, mustard, honey, pickle relish and hot sauce. Set aside.
Thaw the puff pastry until it can be unfolded, about 10-15 minutes. It should still be partially frozen. Cut the puff pastry into 6 even rectangles.
Brush about 1 tablespoon of the ketchup sauce down the center of each puff pastry rectangle. Add the sauted onions as desired.
Place a cube (or tablespoon of shredded) cheese down the center of the puff pastry. Cut slits into the sides of the puff pastry. The slits should be about an inch long and a half inch thick. The center of the puff pastry should be intact - use the cheese as a 'guide' for where not to cut. Cut slits into each end to pull over the ends of the brat.
Next place one of the cooked brats on top of the cheese and pastry. Then in alternating motions, crisscross the puff pastry cuts across the brat to form a 'braid' of sorts. Tuck the ends of the puff pastry under the braid and over the end of the brat. Arrange the brats on a baking sheet lined with parchment.
Brush the brats with whisked egg. Then sprinkle on poppyseeds, sesame seeds, dried minced garlic and onion as desired. Season the brats with salt and pepper, as desired.
Bake the brats for 20-25 minutes, until the brats are golden brown and puffed up. Serve the brats with your favorite BBQ sauce. Enjoy Theon's favorite toy! What is dead may never die.