For shallot butter, mash butter together with chopped shallot, lemon zest, dill, salt, and pepper until evenly incorporated. Shallot butter can be made ahead of time and stored in an airtight container or rolled up in plastic wrap for a few days until needed.
Preheat oven to 400 degrees F. Prepare a large piece of parchment paper, large enough to fold over your piece of fish with a generous 3-4 inches around the edges. If cooking multiple pieces of fish, use a separate piece of parchment for each one. Place parchment on a sheet pan and fold in half, then unfold with one side laid flat on the pan.
Arrange fennel and shallot slices in a layer in the center of one side of the parchment. Lay fish on top of vegetables. Season with salt and pepper. Top with a few pats of shallot butter, slices of lemon, and sprigs of fresh herbs. If using a butterflied fish, place herbs inside of fish cavity.
Drizzle with olive oil.
Fold over the other half of the parchment, then fold and pinch along the edges to seal. You’ll end up with a semi-circle of parchment with crimped edges.
Bake for 15 to 20 minutes until cooked through (adjust time as necessary for thicker/thinner pieces of fish). To check doneness, poke an instant read thermometer into the thickest part of the fish; it should read 145 degrees.
Carefully cut open the packet, avoiding escaping steam, and serve.