Pre-heat the oven to 350°F.
Combine the butter, garlic, shallot, oregano, parsley, lemon zest and juice in a food processor. Pulse to combine.
Pat the turkey or chicken dry and arrange on a cooling rack set over a baking sheet so that the heat circulates around the meat as it roasts. Smear the skin with butter and sprinkle with salt and pepper. Roast for 45 minutes, to an internal temperature of 170°F, until the skin is crisp and brown. Cover with loose foil and rest for 10 minutes before carving.
Meanwhile, cover the potatoes with cold water in a large pot and bring to a boil. Salt the water and cook to tender; drain and return to the pot. Add 2 tablespoons lemon-herb butter, 1/2 cup stock, EVOO and the feta and olives; mash. Season with salt and pepper, to taste. Reserve, covered.
Melt the remaining lemon-herb butter in a skillet over medium heat. Whisk in the flour, cook for 1 minute, then whisk in the remaining stock, about 1 1/2 cups. Cook to thicken, then season with Worcestershire sauce, salt and pepper. Temper the egg yolk with some of the gravy, then return to the sauce to add gloss and deepen the gravy’s smooth finish.
Slice the turkey or chicken and serve with the mashed potatoes and gravy.