Preheat the oven to 425 degrees . Press the dough into the bottom and sides of a 9-inch fluted tart pan with a removable bottom. Pierce the dough with a fork and refrigerate until chilled, about 10 minutes.
Place the tart shell on a baking sheet and fit aluminum foil into it. Fill the shell with pie weights or dried beans and bake for 10 minutes. Remove the weights and the foil and reduce the heat to 375 degrees . Bake until golden and crisp, about 12 minutes more. Let the tart shell cool on a wire rack.
Melt the butter in a medium skillet over medium-high heat. Add the shrimp and cook, stirring, until firm and just opaque, about 2 minutes. Remove from the heat.
In a medium bowl, whisk together the eggs, half-and-half, chives, salt and pepper. Arrange the shrimp, red peppers and feta cheese in the baked shell and cover with the egg mixture.
Bake the quiche until a knife inserted in the center comes out clean, about 25 minutes. Cool for 15 minutes, then carefully remove the side of the pan and transfer the quiche to a serving platter. Serve warm or at room temperature.