With the rack in the middle position, preheat the oven to 400°F (200°C).
Wrap the beets individually in foil. Bake for 1 hour 15 minutes or until tender. Let cool. Peel and cut into slices ¼ inch (½ cm) thick. Set aside.
In a food processor, combine the flour and salt. Add the butter and pulse a few seconds at a time, until the butter is the size of peas. Add the water. Pulse again until the dough just begins to form. Add more water as needed. Remove the dough from the food processor and form into a disc with your hands.
On a floured work surface, roll out the dough. Line a 10-inch (25 cm) quiche plate with a removable bottom with the dough. Refrigerate for 30 minutes.
Place the rack in the lowest position of the oven.
In a bowl, combine 1 cup (100 g) of the feta with the cream, eggs and chives. Season with salt and pepper. Pour into the crust. Top with the beets in a rosette pattern, alternating the colours. Sprinkle with the remaining cheese.
Bake for 45 minutes. Let stand for about 10 minutes before serving.