In a mixing bowl, combine 8 ounces of cream cheese, 1/4 cup of granulated sugar, and 1 teaspoon of vanilla extract. Beat them together until smooth and creamy, about 2-3 minutes. Place small dollops onto a parchment-lined baking sheet and freeze for 20-30 minutes until firm.

In a large bowl, cream together 1/2 cup softened butter, 1/2 cup brown sugar, and 1/4 cup granulated sugar until light and fluffy, about 3-4 minutes. Add in 1 large egg and 1 teaspoon of vanilla extract, blending until fully incorporated.

In a separate bowl, whisk together 1 3/4 cups all-purpose flour, 1/2 teaspoon baking soda, and 1/4 teaspoon salt until well combined. Gradually add the dry mixture into the wet ingredients, stirring just until the dough forms.

If the cookie dough feels too sticky to handle, cover the bowl with plastic wrap and chill it in the refrigerator for 15-30 minutes.

Preheat the oven to 350°F (175°C). Take about 2 tablespoons of cookie dough, flatten it into a disc, and place a frozen cheesecake dollop in the center. Encapsulate the dollop with dough, pinching the edges to seal.

Arrange the assembled cookies on a lined baking sheet and bake for 11-13 minutes, or until the edges are golden brown and the centers are soft. Let cool on the pan for 5 minutes before transferring to wire racks.