Season the flour and sift into a large bowl. Add the butter and rub into the flour using your fingertips until the mixture resembles fine breadcrumbs. Sprinkle with 6 to 7 tablespoons of chilled water (enough to just bind the dough) and draw the mixture together with a knife. Knead gently on a floured board then wrap in plastic wrap and chill for 30 minutes in the fridge.
Preheat the oven to 350°F.
Divide the pastry into 4 equal pieces. Roll each piece on a lightly floured work surface. Roll using short strokes and always in one direction, giving the pastry a quarter turn after each roll. Roll to 1/4in thickness. Cut into a circle about 7in diameter using a plate as a guide.
Mix together the vegetables, lamb and mint. Arrange quarter of the mix in the centre of each pastry circle. Season.
Lightly brush the edge of the pastry with water then carefully bring up the edges to the centre to cover the filling. Pinch the join to create a sealed edge over the top of the pasty. Repeat with the other pastry circles.
Brush the pasties with beaten egg and sprinkled with cracked black pepper. Cook for 1 hour until the pastry is golden and the filling cooked through.