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Heat oven to 180C/160C fan/gas 4. Pulse the flour, oats and butter in a food processor until crumb-like, add the grated carrot and continue to process until the mixture comes together to form a ball. Alternatively rub the butter into the dry ingredients, add the carrots and press together until a ball is formed. Carefully roll out the pastry and line a metal 25cm pizza dish or shallow tart case. Chill.
Gently cook the onions in oil for about 15-20 mins until beginning to caramelise. Place the beetroot in the pastry case and cover with the onions. Whisk the eggs and milk together, season and pour over the onions and beetroot. Bake in the oven, lowering to 160C/140C fan/ gas 3 after 15 mins, for 40 mins or until firm and golden.