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Heat oven to 200C/180C fan/gas 6 and put a baking sheet inside to heat up. Spread each pitta with 1 tsp purée. Top with the tomatoes, shallot, chorizo and cheddar.
Place on the hot sheet and bake for 10 mins until the pittas are crisp, the cheese has melted and the chorizo has frazzled edges. Scatter with basil, if you like, and serve with a green salad.