Nana's Butter Cookies with Milk-Jam Filling

Using an electric mixer on high speed, beat butter, powdered sugar, and vanilla in a medium bowl until light and fluffy, about 4 minutes. Reduce speed to low; add milk, salt, and 2 1/2 cups flour and mix just to combine. Form dough into a 1/2"-thick disk and wrap in plastic. Chill until firm, at least 1 hour.
Let dough sit at room temperature to soften slightly, about 5 minutes. Roll dough out between 2 sheets of parchment paper until about 1/8" thick. Cut out rounds with a floured 2 1/2"-diameter cookie cutter. Cut out a circle in half of the rounds with a 3/4"-diameter cookie cutter and transfer to 2 parchment-lined baking sheets, spacing 1" apart.
Whisk egg and 1 teaspoon water in a small bowl and brush tops of cookies with egg wash. Bake cookies, rotating halfway through, until golden brown, 12&endash;15 minutes. Transfer baking sheets to wire racks and let cookies cool.
Combine milk, sugar, and baking soda in a large saucepan; scrape in vanilla seeds and add pod. Bring just to a boil over medium heat, stirring to dissolve sugar. Reduce heat and simmer, whisking occasionally, until mixture is reduced to about 1 1/4 cups (it will darken and separate, with little bits visible), 80&endash;90 minutes. Strain milk jam through a fine mesh sieve into a medium bowl and let cool.
Spoon a small amount of milk jam onto flat side of a cookie without a hole and sandwich with a cookie with a hole. Repeat with remaining cookies and filling. Dust with powdered sugar, if desired.