- 1 1/2 cups (3 sticks) unsalted butter, room temperature
- 2/3 cup powdered sugar
- 2 teaspoons vanilla extract
- 2 tablespoons whole milk
- 2 teaspoons kosher salt
- 2 1/2 cups all-purpose flour
- 1 large egg
- 1 quart goat's milk or cow's milk
- 1 1/4 cups granulated sugar
- 3/4 teaspoon baking soda
- 1/2 vanilla bean, halved lengthwise
- Powdered sugar (optional; for serving)
- 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.