Place potato in a microwave-safe bowl. Cover with salted water. Cook in the microwave until tender, about 5 minutes.
Place beet tops and stems on a flat work surface. Cut greens into thin strips and dice stems. Stack greens and stems in separate piles.
Heat bacon grease in a large skillet over medium-high heat. Add beet stems, onion, and carrots. Cook and stir until tender, about 5 minutes. Season with salt. Add beet greens; saute until wilted, about 5 minutes.
Reserve 1/2 cup of the greens mixture in the skillet. Transfer the rest of the mixture into a blender; add boiled potato and 2 cups stock. Blend until smooth, adding salty potato water to thin mixture as needed.
Pour the blended mixture into a large pot. Stir in the reserved greens mixture, the remaining 2 cups stock, shredded pork, and peas. Cook until soup is heated through, about 8 minutes. Season with salt and pepper.
