In a medium bowl, beat the cream cheese, 3/4 cup powdered sugar, and 1 tablespoon flour until smooth and creamy.
Line a baking sheet or plate with wax paper and scoop 2-teaspoon mounds of the cream cheese mixture onto the paper.
Place the cream cheese mounds in the freezer for at least 2 hours, or until firm.
In a medium bowl, whisk together 1 2/3 cups flour, cocoa powder, baking soda, and salt; set aside.
In a large bowl, beat the butter, brown sugar, and 1/3 cup granulated sugar until light and creamy.
Add the egg, vanilla bean paste, vinegar, and red food coloring to the butter mixture and beat until well combined.
Add the dry ingredients to the wet ingredients and beat on low speed just until a soft, uniform dough forms.
Cover the bowl tightly with plastic wrap and refrigerate the dough for at least 2 hours or up to 48 hours.
Line 1–2 baking sheets with parchment paper and set aside; keep the cream cheese centers in the freezer.
Scoop about 1 to 1 1/2 tablespoons of cookie dough, flatten it slightly in your hand, and curve the edges up to form a shallow bowl shape.
Remove one cream cheese mound from the freezer and place it in the center of the dough.
Scoop another 1 to 1 1/2 tablespoons of dough, flatten slightly, and place over the cream cheese.
Pinch the edges of the dough together to fully seal in the cream cheese, then roll gently into a ball.
If desired, roll the dough ball in the additional 1/3 cup granulated sugar to coat.
Place the dough ball on the prepared baking sheet and repeat with the remaining dough and cream cheese centers, keeping unused centers in the freezer as you work.
Cover the dough balls lightly and freeze for at least 30 minutes.
When ready to bake, preheat the oven to 350°F and ensure racks are set in the middle position.
Arrange the frozen dough balls on parchment-lined baking sheets, spacing them at least 2 inches apart; do not thaw.
Bake one sheet at a time for 12 to 14 minutes, or until the tops look set and the edges are just firm.
Remove the baking sheet from the oven and cool the cookies completely on the sheet.
For the glaze, whisk together 1 tablespoon milk and 1 cup powdered sugar until smooth and thick but pourable; add a few drops more milk or powdered sugar as needed to reach a drizzle consistency.
Drizzle the glaze over the cooled cookies, using a spoon or piping from a small plastic bag with the corner snipped off.
Let the glaze set before serving or storing the cookies.