Cookies:
Step 1
The use of an electrical mixer on excessive urge, beat butter, powdered sugar, and vanilla in a medium bowl until light and fluffy, about 4 minutes. Lower urge to low; add milk, salt, and a pair of 1/2 cups flour and combine compatible to combine. Invent dough into a 1/2″-thick disk and wrap in plastic. Take a seat again until company, no longer no longer as much as 1 hour.
Step 2
Let dough take a seat at room temperature to melt a puny of, about 5 minutes. Roll dough out between 2 sheets of parchment paper until about 1/8″ thick. Slash out rounds with a floured 2 1/2″-diameter cookie cutter. Slash out a circle in half of the rounds with a 3/4″-diameter cookie cutter and switch to 2 parchment-lined baking sheets, spacing 1″ aside.
Step 3
Lunge egg and 1 teaspoon water in a puny bowl and brush tops of cookies with egg wash. Bake cookies, rotating halfway via, until golden brown, 12&endash;Quarter-hour. Switch baking sheets to wire racks and let cookies cool.
Milk Jam and Assembly:
Step 4
Mix milk, sugar, and baking soda in a tidy saucepan; spot in vanilla seeds and add pod. Bring compatible to a boil over medium heat, stirring to dissolve sugar. Lower heat and simmer, whisking every so continually, until combination is reduced to about 1 1/4 cups (it will darken and separate, with puny bits visible), 80&endash;90 minutes. Stress milk jam via a supreme mesh sieve into a medium bowl and let cool.
Step 5
Spoon a puny quantity of milk jam onto flat aspect of a cookie with out a hole and sandwich with a cookie with a hole. Repeat with ultimate cookies and filling. Mud with powdered sugar, if desired.
