Bring a large pot of salted water to a boil. Once boiling, drop the beets into the water. Bring the beets up to a boil. Lower the heat to a simmer and cook until the beets are tender when pierced with a knife, about 30-40 minutes. Drain the beets and set aside to cool slightly. Once you can handle them, slip the skins off of the beets. Set the peeled beets aside.
While the beets are simmering, make the pesto. In a food processor fitted with the “S” blade, place the garlic, capers, and pistachios. Pulse these items a couple times until finely chopped. Then, to the food processor, add the mint, parsley, lemon juice, miso, salt, and pepper (being careful on the salt because the miso and capers are quite salty). Pulse this mixture until the herbs are finely chopped. Then, with the motor on low, drizzle the olive oil in through the feed tube until you have a slightly fluid paste. Scrape down the sides of the food processor bowl and pulse a couple more times. Check the pesto for seasoning, adjust if necessary, and set aside.
Preheat a grill to high. Cut the beets into halves (or into quarters if you like and don’t mind fussing with them on the grill). Place the cut beets in a medium bowl and toss them with olive oil, the smoked paprika, salt, and pepper. Once evenly coated, set aside.
Using tongs, carefully place the beets on your preheated grill. Let them sit for about 4-5 minutes, or until there is some slight charring on the side facing the grill. Flip the pieces of beet over and grill for another 4-5 minutes to get slight charring on the other side. Remove the beets from the grill.
To serve, spread the pesto out evenly on a dinner plate-sized platter. Arrange the grilled smoky beets on top. Top the beets with the micro greens, the crumbled vegan cheese (if using), some extra mint, a few twists of black pepper, and some extra capers if you like. Serve warm or at room temperature.