In a medium bowl, beat the cream cheese, 3/4 cup powdered sugar, and 1 tablespoon flour until smooth and creamy.
Line a cookie sheet or plate with wax paper and scoop 2-teaspoon mounds of the cream cheese mixture onto the lined surface.
Place the tray in the freezer and freeze the cream cheese scoops for at least 2 hours, or until very firm.
In another medium bowl, whisk together 1 2/3 cups flour, cocoa powder, baking soda, and salt until well combined; set aside.
In a large bowl, beat the softened 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 fully combined.
Add the dry ingredients to the wet ingredients and beat on low speed just until no dry streaks remain.
Cover the bowl tightly with plastic wrap and refrigerate the cookie dough for at least 2 hours or up to 48 hours, until firm.
Once the dough is chilled and the cream cheese scoops are frozen, line baking sheets with parchment paper and set aside.
Scoop about 1 to 1 1/2 tablespoons of cookie dough and roll it into a ball, then flatten it slightly and press the center to create a shallow bowl shape.
Working with one cream cheese scoop at a time, remove a frozen scoop from the freezer and place it in the center of the dough.
Scoop another 1 to 1 1/2 tablespoons of dough, flatten it slightly, and place it on top of the cream cheese filling.
Pinch the edges of the top and bottom dough pieces together to fully seal in the cream cheese, then gently roll into a smooth ball, making sure no filling shows.
If desired, roll each dough ball in the additional 1/3 cup granulated sugar to coat the outside.
Place the filled dough balls on a parchment-lined tray, cover lightly, and freeze for at least 30 minutes, or until firm.
When ready to bake, preheat the oven to 350°F and arrange the frozen dough balls on prepared baking sheets, spacing them at least 2 inches apart; do not thaw.
Bake one sheet at a time in the center of the oven for 12–14 minutes, or until the tops look set and the edges are just firm.
Allow the cookies to cool completely on the baking sheet so the centers can set before glazing.
For the glaze, whisk 1 tablespoon milk with 1 cup powdered sugar until smooth, then add more milk or powdered sugar as needed until the glaze is white, smooth, and thin enough to drizzle.
Drizzle the glaze over the cooled cookies using a spoon or by transferring it to a small bag with the tip snipped off, then let the glaze set before serving.