In a skillet over medium heat, cook the breakfast sausage, breaking it up with a spoon, until fully browned and cooked through, about 6-8 minutes. Drain any excess fat and transfer to a bowl. Let cool slightly, then mix in the shredded cheese, parsley, thyme, and black pepper.

Preheat oven to 375°F (190°C). Line a baking sheet with parchment paper. On a lightly floured surface, roll the dough into a 12x16-inch rectangle. Brush lightly with olive oil. Spread the sausage and cheese filling evenly over the dough, leaving a 1/2-inch border along one long edge.

Starting from the long edge covered with filling, tightly roll the dough into a log, pinching the clean edge to seal. With a sharp serrated knife, slice the log into 12 equal pieces (about 1 1/4-inch thick).

Place the rolls cut-side up on the prepared baking sheet, spacing them about 1 inch apart. In a small bowl, whisk the egg with 1 tbsp water. Brush the tops and sides of each roll generously with the egg wash. Sprinkle with sesame seeds if using.

Bake for 22-25 minutes, or until the rolls are puffed, deeply golden brown, and the filling is hot and bubbly. Rotate the tray halfway through for even browning.

Let cool on the baking sheet for 5 minutes before serving warm. Arrange on the tray or transfer to a platter.