In a medium bowl, beat the cream cheese, powdered sugar, and vanilla extract until smooth and creamy. Scoop the mixture into 16 small portions on a parchment-lined baking sheet. Freeze for 45 minutes to 1 hour, or until firm to the touch.

Preheat the oven to 350°F (175°C) and line baking sheets with parchment paper. In a large bowl, cream the butter, brown sugar, and granulated sugar together until light and fluffy. Beat in the eggs, one at a time, followed by the vanilla extract and red food coloring.

In a separate bowl, whisk together the flour, cocoa powder, cornstarch, baking soda, and salt. Gradually fold the dry ingredients into the wet mixture until just combined. Gently fold in the white chocolate chips.

Divide the cookie dough into 16 equal balls and flatten each into a disc using your hands. Place one frozen cheesecake portion in the center of each disc, ensuring the cheesecake is cold to prevent stickiness. Carefully wrap the dough around the cheesecake, pinching seams and rolling gently into a smooth ball to fully encase the filling and avoid leaks.

Place the assembled cookie balls on the prepared baking sheets, spacing them about 2 inches apart. Bake for 10-13 minutes until the edges are set but centers still look soft. Remove cookies from the oven and immediately press down gently with a round spatula, the rim of a glass, or a spoon to shape them into perfect circles. Allow the cookies to cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.

For a gourmet touch, drizzle melted white chocolate over the cooled cookies or sprinkle with festive sprinkles.