In a mixing bowl, combine shredded cheese, cream cheese, minced garlic, fresh thyme leaves, salt, and black pepper until evenly mixed.
Preheat oven to 400°F (200°C) and line a baking sheet with parchment paper.
Lightly flour a surface and slightly roll out each sheet of puff pastry.
Spread the cheese mixture evenly over both pastry sheets.
Roll each sheet tightly into a log shape, then cut into 2–3 inch rectangular rolls.
Place the rolls seam-side down on the prepared baking sheet.
Beat the egg and brush over the tops of the rolls for a golden finish.
Optionally brush tops with olive oil and sprinkle extra fresh thyme.
Bake in the preheated oven for 20–25 minutes or until pastries are puffed and golden brown.
While baking, warm marinara sauce in a small saucepan over medium heat.
Stir in dried oregano, red pepper flakes if using, and olive oil. Simmer for 3–5 minutes.
Remove rolls from oven and serve warm with the red dipping sauce.