Step 1 Preheat oven to 190°C. Make pastry: Combine flour and baking powder in a food processor on low speed. Add olive oil slowly until well combined. Add 60ml cold water, a little at a time, until mixture clumps together. Transfer dough to a floured work surface.
Step 2 Flour hands and form dough into a ball; knead gently until just smooth, then flatten slightly into a disc. Wrap disc in cling film and chill in fridge for 10–15 minutes.
Step 3 Meanwhile, make filling: Heat olive oil in a large non-stick frying pan set over medium–high heat. Add onion and garlic; sauté for 2 minutes. Reduce heat to medium. Stir in spices; cook for 1 minute. Add mushrooms and carrot with 1 cup water; simmer for 5 minutes. Add remaining filling ingredients and simmer for another 20 minutes. Remove pan from heat and leave to cool.
Step 4 Transfer chilled pastry disc to a clean work surface and roll out into a large round to cover pie.
Step 5 Spoon filling into a 22.5cm pie dish and cover with pastry round, trimming to fit. Score pastry with a sharp knife, making 2 or 3 small slits in the centre (to let steam escape); brush with olive oil.
Step 6 Transfer dish to oven; cook for 45–50 minutes, or until golden. Remove pie from oven, leave to cool for 5 minutes and serve.