In a large bowl, mix together the turkey, shallot, garlic, seasonings, milk, breadcrumbs, Worcestershire sauce, and mayonnaise. Mix until well combined.
Form the mixture into ½-inch thick patties and place on a baking sheet. Bake for 20 minutes, or until internal temperature is between 155-160 F.
If cooking on the stovetop, heat a thin layer of oil in a skillet over medium heat. Once hot, add patties, working in batches as needed. Cook until deeply browned on each side, about 10 minutes total.
Once cooked, place patty on arugula-lined bun and top with feta cheese, or serve as desired.