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Heat oven to 200C/180C fan/gas 6. Roll out the pastry on a lightly floured surface until large enough to line a 28cm loose-bottomed tart tin. Make sure the pastry is pressed snugly into the tin. Leave the remaining pastry overhanging the sides. Lift onto a baking sheet. Line the pastry case with baking parchment and baking beans, then bake blind for 25 mins. Remove the paper and baking beans, and return to the oven for 5 mins more to dry out the pastry on the base.
While the pastry is baking, boil the broad beans and peas for 5 mins until just tender. Drain and rinse off any starchy froth that appears. Tip into a bowl with the basil, 3 tbsp cream and plenty of seasoning. Purée with a hand blender (or in a food processor) until smooth. Dry-fry the bacon for a few mins in a non-stick pan until starting to turn golden.
Beat the remaining cream with the eggs, milk, Parmesan and seasoning. Smooth the pea and bean purée over the base of the tart case and scatter with the bacon. Carefully pour on the egg mixture. Turn down oven to 180C/160C fan/gas 4 and bake for 35 mins until set and golden. Cool a little, then carefully trim off the overhanging pastry with a sharp knife. Scatter with extra basil and serve with a salad for a picnic, summer buffet or lunch, or on its own as a starter.