Set oven at 375 degrees. Have on hand a large rimmed baking sheet lined with aluminum foil and sprayed with Pam.
Cut parchment into four 15”x15” squares. Lay them on the counter. Place a small handful of greens in the center of each one and add a sprinkle of dill. Set a piece of fish on top and sprinkle with salt & pepper. Add 6 tomatoes halves and a few slices of shallot to each piece of fish, then add a pad of butter, 2 slices of lemon and a sprinkle of olive oil.
Seal the packets by bringing two of the sides together into a 20 inch strip, continuing to fold toward the center until you have a seam running down the middle of the packet. Lift the packet up and fold the right and left sides underneath.
Place the packets on a baking sheet. Brush the tops with vegetable oil. Bake for 20-30 minutes or until the fish feels firm to the touch and is cooked through.