In a deep, heavy-bottomed saucepan, add the unopened can of sweetened condensed milk and enough cold water to cover the can by 2 inches. Bring to a boil over high heat, then lower the heat to maintain a simmer for 3 hours, checking the pot every 30 minutes or so and adding more water as needed to keep the can submerged (this is very important). Use tongs to remove the can from the pot and set it aside to cool completely at room temperature (this will take several hours; do not open the can while it is hot).
Meanwhile, make the vanilla sugar: In the bowl of a small food processor or spice grinder, add ¼ cup of the sugar and the vanilla bean and seeds; run the machine until the bean is completely incorporated into the sugar, about 20 seconds. Transfer to a small shallow bowl and set aside.
Make the cookies: Line 2–4 large baking sheets with parchment paper and set them aside.
In a stand mixer fitted with the paddle attachment, add the butter and remaining ¾ cup sugar and mix on medium speed until smooth and well blended, 3–4 minutes. Add the egg yolk, milk, and vanilla and continue mixing for 10–15 seconds (the mixture may appear curdled or broken). In a medium bowl, whisk the flour, baking powder, and salt. Add the flour mixture to the butter mixture, turn the mixer on low speed, and mix just until a homogenous dough comes together, about 1 minute.
Scoop out 1-tablespoon portions of dough and roll each into a smooth ball: dredge in the vanilla sugar to coat, then transfer to the prepared baking sheets (leave at least two inches between each cookie). Flatten the balls slightly, then press your thumb or the handle end of a wooden spoon into the center of each one to make a 1-inch-wide depression to hold to the filling. (There will be some small cracks in the edges; gently press any large cracks back together). Transfer the baking sheets to the freezer and freeze until the cookies are firm (about 30 minutes).
Meanwhile, set a rack in the center of the oven and preheat to 350°F. Bake one sheet of cookies at a time until they are lightly golden at the edges but the tops remain pale and soft, 10–12 minutes. Remove the sheet and, using the handle end of a wooden spoon or whisk, press down the center of the cookie once more. Let the cookies cool on the baking sheet for 5 minutes, then transfer to a wire rack to finish cooling while you continue baking the rest.
Once all of the cookies are baked and cooled, transfer the powdered sugar to a shaker or fine mesh sieve; sprinkle the tops of the cookies generously with powdered sugar. Use a piping bag or a spoon to fill the well of each cookie with about 1 teaspoon of the prepared dulce de leche. Serve immediately or store in a single layer in an airtight container for up to 3 days.