Set oven to 425°F. Prepare a baking sheet by lining it with parchment paper or silicone baking sheet liner. Set aside.
Using a fork, whisk 1 large egg together with 1 teaspoon water in a small bowl. Set aside.
Unfold or roll out the puff pastry on a lightly floured countertop. Gently roll out the sheet so that it measures 10 inches square. Leaving a 1/2-inch border along the top edge of the pastry, spread 2 Tablespoons gochujang evenly over the pastry (spread it all the way to the remaining 3 edges). Sprinkle 1 cup shredded sharp cheddar cheese and the thinly sliced green onion evenly over the gochujang, avoiding the 1/2-inch border at the top.
Starting at the bottom edge (the edge closest to you), carefully roll up the pastry into a log (use a bench scraper to help you roll it up if the pastry sticks to the counter). Once you get to the top, lightly brush the 1/2-inch border with the egg mixture. Gently press the seam closed with your fingers.
If you plan to bake the pinwheels right away, cut the log into 12 equal pieces with a serrated knife, being careful not to squish the pastry too much. Place the pieces on the baking sheet, leaving at least 1 inch between the pieces, reshaping them to circles if needed. Gently brush the tops and sides of the spirals with the egg mixture making sure not to leave large puddles of egg around the pieces. Sprinkle lightly with sesame seeds if desired.
Bake until the cheese is melted and bubbling and the pastry is golden brown, about 13-17 minutes. Cool for 5 minutes before serving.
Notes
If you choose to double the recipe (it is easy to double if you have two sheets of puff pastry), don't double the egg wash part of the recipe (one egg and a teaspoon of water). You will have lots of egg wash leftover if you only make a single recipe.