Scrape the carrots and cook for 3 minutes.. Top and tail the haricots verts and cook them for 2 minutes. Drain and rinse under cold running water to stop the cooking process.
Grease the cake tin with butter. Line the cake tin with 6 slices of puff pastry. Trim the edges for a neat fit, if necessary. Arrange half the carrots lengthwise in the tin. Now make a layer with half the haricots verts. Repeat with the carrots and haricot verts to make 4 layers.
Thinly slice the garlic. Beat 8 eggs with the garlic, cream, flour, Herbes de Provence with the mixer until smooth. Season with salt and pepper. Pour the mixture evenly over the vegetables in the cake tin. Cover carefully with the rest of the puff pastry. Press the edges well together and trim if necessary. Beat the egg and brush the top of the pastry with the egg.
Bake the terrine in the middle of the oven for about 50 minutes until golden brown and cooked. Cool the terrine in the tin on a wire rack. When cool, remove from the cake tin and cut into slices.
