For the dough, in a small bowl, combine the water, yeast, sugar and olive oil. Let stand until foamy, about 5 minutes.
In the bowl of a food processor, pulse both the flours and salt. Add the yeast mixture and process until a ball of dough forms. Transfer the dough to an oiled bowl. Cover and let rise until doubled in volume, about 2 hours.
Preheat the oven to 400 degrees F with a pizza stone inside the oven.
Place each beet on an individual sheet of aluminum foil. Drizzle the beets with olive oil and season with salt and pepper. Fold into loose packets and place on the pizza stone. Roast the beets for 45-50 minutes until they are easily pierced with a knife.
When the beets are cool enough to handle, use a paper towel to rub off the skin. Cut into 1/8-inch-thick slices and set aside.
For the pesto, pulse the garlic cloves in the bowl of a food processor fitted with the metal blade until they are minced. Add the beet greens, walnuts, Parmesan, salt and pepper. Puree the mixture and while the motor is running, drizzle the olive oil through the feeder tube until the pesto is fully combined.
Divide the dough in half. Roll out each one into an oval. Bake on the preheated pizza stone for 10-12 minutes until light golden brown.
Spread the pesto on the flatbreads leaving a 1-inch border at the edges. Arrange the sliced beets on top.
Toss the arugula with the lemon juice in a small bowl and scatter on the flatbreads. Sprinkle with goat cheese, walnuts and chives.
Store in extra pesto in an airtight container in the refrigerator.