Wash and peel the turnip. Put it into a medium pot, cover it with water, and add a pinch of salt. Bring it to a boil and cook until tender, about 15 to 20 minutes; set aside. While the turnip is cooking, wash and peel the beets. Remove the greens, wash them well, and set aside. Put the beets into the boiling water and cook until tender, about 20 to 30 minutes. Remove and set them aside.
Slice the papaya in half, scoop out and discard the seeds. Carefully scoop out the fruit and reserve the shells. Coarsely chop the papaya and put it into a small bowl.
Melt 2 tablespoons butter in 1 tablespoon olive oil in a medium skillet over medium-high heat. Add the beet greens and 1/2 the garlic. Cook the beet greens until wilted, about 5 to 7 minutes; season with salt and pepper. Remove and set aside. Add the remaining butter, olive oil, and garlic to the pan and cook the shrimp until they are pink and done, about 3 minutes; season with salt and pepper. Add the shrimp to the papaya and mix well.
Fill the reserved papaya shells with the shrimp mixture and put them onto plates. Serve with slices of turnips and beets, the beet greens, and slices of tomato.