Trim woody ends off asparagus and cut into 1" lengths on an angle. Mince garlic. Cut mushrooms into 1/4" slices. Peel and slice shallot into 1/4" rounds. Separate rounds into rings. Halve grape tomatoes. Stem and coarsely chop oregano. Place asparagus, corn, garlic, 1 Tbsp. olive oil, 1/2 tsp. salt, and 1/4 tsp. pepper in a medium mixing bowl and toss to combine.

Pat cod fillets dry, and season both sides with 1/4 tsp. salt and 1/4 tsp. pepper. Place a medium non-stick pan over medium heat. Add 2 tsp. olive oil and cod to hot pan and cook undisturbed until lightly browned on one side, 2-4 minutes. Transfer cod to a plate. Cod will finish cooking in a later step. Reserve pan; no need to wipe clean.

Return pan to medium-high heat and add 1 tsp. olive oil and mushrooms to hot pan. Cook until lightly browned, 3-5 minutes. Add shallots and cook 1 minute. Remove from burner. Fold parchment paper in half widthwise and cut on fold. Fold paper again like a book. Open paper and place asparagus-corn mixture in center of one half of each paper, leaving a 21/2" to 3" border. Add mushroom-shallot mixture to each pile and top with cod. Repeat with second paper. Reserve mixing bowl.

Fold paper over, covering mixture. Starting at folded edge, fold lower left corner at an angle up toward mixture. Start from middle of previous fold and fold again toward mixture. Repeat process around mixture, making as tight a seal as possible. Seal with a small fold under packet.

Place packets on prepared baking sheet and roast until packets lightly brown on edges and cod reaches a minimum internal temperature of 145 degrees, 6-8 minutes. While cod bakes, stir grape tomatoes, oregano, 1 Tbsp. olive oil, 1/4 tsp. salt, and a pinch of pepper in reserved mixing bowl.

Traditionally, the packet is cut open at the table; however, feel free to remove cod and vegetables from packet and serve on a plate if desired. Spoon relish on top of cod and drizzle with 1/2 tsp. olive oil.