In a medium bowl, beat the cream cheese, 3/4 cup powdered sugar, and 1 tbsp flour until smooth and creamy.
Line a small baking sheet or large plate with wax paper. Scoop the cream cheese mixture into 2-teaspoon mounds onto the lined surface.
Freeze the cream cheese mounds 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 softened butter, brown sugar, and 1/3 cup granulated sugar until creamy and slightly fluffy.
Add the egg, vanilla bean paste, vinegar, and red gel 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 dough forms and no dry streaks remain.
Cover the dough tightly with plastic wrap and refrigerate for at least 2 hours or up to 48 hours.
Line a baking sheet with parchment paper. Scoop about 1 to 1 1/2 tbsp of chilled dough, flatten it slightly in your hand, and gently curve the edges up to form a shallow cup.
Take 1 frozen cream cheese mound from the freezer and place it in the center of the dough cup, keeping the remaining mounds frozen as you work.
Scoop another 1 to 1 1/2 tbsp of dough, flatten it slightly, and place it over the cream cheese mound.
Pinch the edges of the dough together to fully seal in the filling, then roll gently between your palms to form a smooth ball.
If rolling in sugar, place 1/3 cup granulated sugar on a small plate and roll each dough ball in the sugar to coat.
Place the dough balls on the parchment-lined baking sheet, cover, and freeze for at least 30 minutes or until firm.
When ready to bake, preheat the oven to 350°F and line additional baking sheets with parchment paper.
Arrange the frozen cookie dough balls on the prepared baking sheets, spacing them at least 2 inches apart. Do not thaw before baking.
Bake one sheet at a time in the center of the oven for 12 to 14 minutes, or until the tops look set and the edges are slightly firm.
Allow the cookies to cool completely on the baking sheet so the cream cheese centers can set.
For the glaze, whisk together 1 tbsp milk and 1 cup of powdered sugar in a small bowl until smooth, then add up to 1/4 cup more powdered sugar as needed until the glaze is white and thin enough to drizzle.
Drizzle the glaze over the cooled cookies, using a spoon or piping it from a small freezer bag with the corner snipped off. Let the glaze set before serving.