With the rack in the middle position, preheat the oven to 200 °C (400 °F).
Wrap the beets individually in aluminum foil. Bake for about 1 hour and 15 minutes or until tender. Let cool. Peel and cut into ½-cm (1/4-inch) thick slices. Set aside.
In the bowl of a food processor, combine the flour and salt. Add the butter and pulse for 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 water, if needed. Remove the dough from the food processor and form into a disk with your hands.
On a floured work surface, roll out the dough. Line a 25-cm (10-inch) quiche plate with a
removable bottom. Refrigerate for 30 minutes.
Place the rack in the lowest position of the oven.
In a bowl, combine 250 ml (1 cup) of feta cheese with the cream, eggs, and chives. Season with salt and pepper. Pour into the crust. Place the beets in a rosette shape, alternating colours. Sprinkle with the remaining cheese.
Bake for about 45 minutes. Let stand about for 10 minutes before serving.