Use a large baking tray. Grease with oil. Roll out dough to fit baking tray. Place dough on tray.
Heat 2 tbsp olive oil in a pan; fry the onion and garlic until transparent. Add tomatoes; cook, uncovered, for 5 minutes to reduce. Stir in the tomato puree and season with oregano and salt and pepper.
Preheat oven to 250°C. Clean mushrooms with a paper towel, if needed; slice finely. Cut ham into strips. Remove rind from gouda and grate coarsely; slice the mozzarella.
Spread the tomato mixture over dough. Scatter the mushrooms, ham, both cheeses and the parsley over the top. Season with salt and pepper and drizzle with remaining olive oil. Bake for 12–15 minutes until the edges of the dough are light brown.