Warm 1 tablespoon of the olive oil in a 10-inch skillet over medium-high heat.
Add the onions and red bell peppers and cook for 3 minutes, stirring occasionally, until they soften.
Add the garlic and cook for 30 seconds, until fragrant.
Stir in the tomato paste and cook for 1 minute, until it coats the vegetables evenly.
Add the ground beef, water, ½ teaspoon of the kosher salt, and the black pepper. Cook for 4 minutes, stirring frequently and breaking the beef into small crumbles, until no pink remains.
Transfer the beef mixture to a plate and wipe the skillet clean.
Reduce the heat to medium and return the skillet to the burner.
Brush both sides of the pita with the remaining 1 teaspoon of olive oil and sprinkle with a pinch of the kosher salt.
Toast the pita for 45 seconds per side, until lightly golden.
Transfer the pita to a rimmed baking sheet and spread the shredded gouda evenly over it.
Spoon the beef mixture over the cheese.
Bake for 5 to 6 minutes, until the cheese is bubbling across the surface.
Slice into wedges and top with sliced avocado, if using.
Notes
Use a 10-inch skillet: The pita needs to sit flat against the bottom of the pan to brown evenly. In a smaller skillet the edges ride up the sloped sides and never touch the heat.Shred the gouda yourself: Pre-shredded cheese is coated with starch or cellulose so the pieces don't stick together in the bag, and that coating keeps it from melting into a smooth layer. Since the melted gouda is what seals the crust here, grate it off a block.Break the beef into small crumbles: Stir frequently and break the meat apart as it cooks. Small crumbles spread evenly over the cheese and give you beef across the whole pita. Large clumps leave bare spots.Let the gouda tell you when it's done: Five to six minutes at 400°F is the range, but ovens vary. Pull it when the cheese is bubbling across the surface. The pita is already toasted by then, so the bake is only there to melt the cheese and warm the beef through.