Position a rack in the center of the oven and preheat to 450 degrees . In a bowl, whisk the egg yolk, mustard and lemon juice; season with salt. Whisk in the grapeseed oil in a steady stream until it thickens into mayo. Stir in the tarragon. In another bowl, mix the panko and cheese.
On a rimmed baking sheet or baking dish, scatter the green beans, mushrooms and shallots. Drizzle with EVOO; season with salt and pepper.
Slather the mayo on the seasoned chicken or fish and coat with the panko-parm mixture. Arrange the chicken or fish on the vegetables. Roast until cooked through, about 20 minutes for chicken or 15 minutes for fish.