Bring a large pot of salted water to a boil. Prepare a bowl of ice water. Strip beet greens, turnip greens, and kale from their stems and clean them. Cook each batch in boiling water for about 1 minute, transferring to ice water afterward. Discard the cooking water.

Remove greens from ice water and squeeze out excess water.

Combine the greens, 2/3 cup olive oil, garlic, Parmigiano Reggiano, and kosher salt in a blender or food processor. Blend until smooth to make the pesto and store it in a bowl with plastic wrap.

Heat 1/4 cup olive oil in a large pot over medium-high heat. Add onion, grated tomato, garlic, rosemary, kosher salt, bay leaf, and chili flakes. Cook for 5 minutes, then reduce heat and cook for 20-30 minutes until vegetables turn a deep golden brown.

Add remaining olive oil to the onion mixture and stir in leek, carrot, turnips, and beets. Cook over medium-high heat for 5 minutes.

Pour white wine over the mixture, deglazing the bottom of the pot. Add cannellini beans, green beans, and water. Bring to a boil, then simmer until green beans are tender for about 5 minutes. Stir in parsley and lemon juice.