Preheat the oven to 400°F and line a large baking sheet with parchment paper.
In a mixing bowl, combine the sausage, cheddar cheese, breadcrumbs, 1 egg, and fresh thyme.
In a separate small bowl, whisk together the remaining egg and 1 tablespoon of water until well combined. Set aside.
Unroll the puff pastry on a floured surface. Use a rolling pin to roll it out slightly to create a large rectangle and soften the puff pastry slightly (this helps make it easier to form and shape). Cut the pastry in half to create two long strips. Repeat with the second puff pastry dough, so that there are 4, long equal strips.
Divide the sausage mixture into fourths, shaping each portion into a long log. Place each sausage log in the center, lengthwise along the pastry strips, pressing it down as needed so that it creates a long, thin line of sausage down the center of each puff pastry strip.
Brush one side of the puff pastry with the Dijon mustard, then brush the other side with the egg wash. Roll the pastry around the sausage, letting the egg wash act as glue, pushing the puff pastry together until it seals.
Flip it over so that the seam is on the bottom. Using a paring knife, score small slits into the top of the sausage logs, then cut them into 1½-inch rolls.
Place each roll seam-side down on the prepared baking sheet. Brush the tops with the remaining egg wash and sprinkle with Everything But The Bagel seasoning. Repeat with the remaining dough and sausage.
Transfer to the oven and bake for 18 to 20 minutes, or until the pastry is puffed and golden brown and the sausage is cooked through.
Let the sausage rolls cool slightly before transferring to a platter. Drizzle each sausage roll with hot honey, sprinkle with extra thyme, and serve warm.