Prepare the filling: Preheat the oven to 375°F (190°C) and place an oven rack in the lowest position in the oven. If using fresh cherries, wash, stem, and pit them.
Roll the dough out to line a lightly buttered 9 by 1½-inch round tart pan (with a removable bottom) or ceramic pie plate. Because the filling is liquid, be sure the dough is well pressed into the corners of the mold and free of any cracks or holes (you don't want the filling to leak). Chill the crust until firm, at least 1 hour.
If using frozen or jarred, let defrost and then pat them dry on a paper towel. Rinse the blueberries (and/or raspberries) and set aside to dry.
In a small saucepan, bring the milk, cream, half the sugar, and the vanilla to a boil. Set aside. In a bowl, whisk together the eggs with the remaining sugar, then whisk in the flour. Pour the hot cream mixture over the egg mixture, whisking until smooth, and strain the mixture.
Assembly and baking: Line the entire surface of the refrigerated tart shell with parchment paper or foil and fill with baking weights or beans right to the top (this will help prevent the crust from shrinking as it bakes).
Place the tart shell on a sheet pan and bake for about 15 to 20 minutes. When the shell begins to colour around the sides, remove it from the oven and carefully remove the paper and weights or beans. Bake the shell for a further 5 minutes to allow the base to cook through.
Fill the hot tart shell with the fruit, spreading it out to the edges, and pour over enough filling to reach halfway up the sides of the shell. Place it in the oven and then carefully pour in the rest of the filling so it reaches the top. Be careful not to add too much, or the filling could seep between the pastry and the tart pan.
Bake for 10 minutes at 375°F, then reduce the heat to 350°F and bake for about 25 to 30 minutes more. The clafoutis is cooked when the filling just begins to rise at the sides and sets in the middle.
Let cool on a wire rack and unmold when warm (unless it was made in a pie pan, in which case the clafoutis should be served unmolded). Sprinkle over a pinch of granulated sugar if you like before serving.
For a clafoutis without a crust: Generously butter a 9 by 1½-inch deep porcelain quiche mold or glass pie plate. Reduce the amount of cherries to 1 cup and increase the amount of flour to ¼ cup.