Make the Stock: Preheat oven to 375 degrees F and adjust oven rack to lowest position. Place the turkey wings, celery, onion, and garlic in a large Dutch oven. Roast in the preheated oven, uncovered, for 2 hours. Remove pot from the oven.
Add the water to the turkey and vegetables and bring to a boil over high heat. Reduce heat to low and simmer, uncovered, for 1 hour.
Strain the stock into a bowl with at least 1-quart capacity. (You can eat the turkey wings and discard the used vegetables.) Refrigerate for at least 2 hours or overnight. Remove the accumulated fat on the top of the stock before using.
Make the Gravy: In a 12-inch pan, melt the butter over medium heat. Whisk in the flour and cook for 1 to 2 minutes, until you detect a nutty aroma. Gradually whisk in the defatted turkey stock and continue to cook, whisking frequently, until thickened (it should coat the back of a spoon), about 5 minutes. Whisk in the milk, salt, and pepper. Taste and adjust seasonings as needed. Serve warm.
Notes
Turkey Wings: These are usually widely available in most grocery stores. If you cannot find them, turkey legs can be substituted.
Butter: If you prefer, you can use the fat that rises to the top of the chilled stock in place of the butter when you make the gravy.
Make-Ahead Stock: You can make the stock and keep it in the refrigerator for up to 5 days before preparing the gravy.
Freezing Stock: You can also freeze it for up to 3 months. Thaw in the refrigerator overnight before using.
Make-Ahead Gravy: You can fully prepare the gravy 1 day before serving. Store it in a covered container in the refrigerator. Rewarm in a saucepan over low heat, whisking frequently, until warmed through. Thin with additional milk if the gravy is too thick.
Freezing Gravy: The fully prepared gravy can be frozen in an airtight container for up to 3 months. Thaw in the refrigerator overnight, then reheat on the stovetop according to the directions above.