Make the milk jam: Whisk the milk and granulated sugar in a large microwave-safe bowl until dissolved. (The bowl should be at least 3 quarts to prevent the milk from overflowing.) Microwave 10 minutes, then stop and whisk. Microwave 10 more minutes, whisking halfway through. The mixture should be thick, bubbling and dark ivory in color. If it is still loose, continue microwaving in 1-minute intervals until thickened. Whisk in the vanilla until smooth. You should have about 1/2 cup; the mixture will continue to thicken as it cools.
Make the cookies: Whisk the flour, cinnamon, baking powder and salt in a medium bowl. Beat the butter, granulated sugar, brown sugar and ground coffee in a large bowl with a mixer on medium-high speed until fluffy, 3 minutes. Beat in the egg and vanilla until combined. Reduce the mixer speed to low and beat in the flour mixture in two batches until just combined.
Roll heaping tablespoonfuls of dough into 1-inch balls. (If the dough is too soft to roll, refrigerate about 20 minutes.) Dip the tops of the dough balls in the turbinado sugar, then make a 1/2-inch indentation in the center of each with the handle of a wooden spoon. Place on a large plate and refrigerate until firm, at least 30 minutes.
Position racks in the upper and lower thirds of the oven and preheat to 350˚ F. Line 2 baking sheets with parchment paper. Arrange the dough balls about 2 inches apart on the pans. Bake, switching the pans halfway through, until the cookies are set around the edges (but not browned) and the centers are still soft, 12 to 14 minutes. Remove from the oven and re-indent the center of each cookie with a 1/2-teaspoon measuring spoon. Fill each indentation with 1/2 teaspoon of the milk jam. Transfer the cookies to a rack to cool completely.