Set up a grill for cooking with indirect heat. If using a charcoal grill, build the coals on one side only. If using a gas grill, heat one side only. Place a heavy-bottomed enameled cast-iron pan over the direct heat side of the grill and add the olive oil.
Add the onions, fennel and garlic with a pinch of salt and allow to begin to soften, 2 to 3 minutes. Stir in the tomato paste and cook until rusty, 2 to 3 minutes.
Deglaze the pan with 1 cup water, scraping the bottom for any brown bits. Add all of the greens and season with salt and pepper.
Cover the pan with a lid.
Move the pan to the indirect-heat side of the grill, then close the lid and cook until the greens are wilted, 5 to 7 minutes. Remove from the grill and stir in the dill and lemon juice. Serve.