Preheat the oven to 425°F. Line a rimmed baking sheet with parchment paper.
Peel the beet and cut it into 1- to 2-inch cubes. Place the beet cubes in a medium bowl and add 2 tablespoons of the oil, the salt, and 5 grinds of black pepper. Toss well to coat all the pieces, then dump them onto the prepared baking sheet and spread them out in a single layer.
Place the baking sheet in the oven, and roast the beet cubes for 40 minutes or until they are easily pierced with a fork. Remove the baking sheet from the oven.
While the beet roasts, put the pine nuts or walnuts in a small, dry skillet over high heat and toast until they are fragrant and browned in spots.
In a food processor, combine the roasted beets, toasted nuts, garlic, chives, cheese, and lemon juice. Pulse until everything is very finely chopped, then add the remaining ¼ cup of olive oil and process until the mixture resembles a thick, mostly smooth paste.
Taste and adjust seasonings as desired. Spread ¾ to 1 cup of the pesto on a 12- to 14-inch round of pizza dough, add other toppings, and bake.
Or, if you're not making pizza, enjoy the pesto as a snack with crackers or stirred into hot cooked pasta as a yummy vegetarian dinner.