1. Prepare the Cheesecake Filling: Beat 3 to 8 ounces of cream cheese with 1 1/2 tablespoons to 1/4 cup sugar and 1/4 to 1 teaspoon vanilla extract until smooth, about 2-3 minutes. Chill the filling, either by freezing small dollops or refrigerating a piping bag, until firm (30 minutes to 1 hour).
2. Make Brown Butter (if using): Heat 1/2 cup unsalted butter cubes over medium heat, stirring constantly until golden brown and foamy. Cool before use. Alternatively, use softened butter for a traditional creaming method.
3. Prepare the Dough: Whisk cooled brown butter with brown and granulated sugars. Add the egg(s) and vanilla, mixing until combined. Sift together flour, baking soda (or baking powder), cinnamon, and salt. Fold dry ingredients into wet mixture until just combined. Refrigerate the dough briefly (about 20 minutes) if too soft but avoid over chilling to prevent hard dough.
4. Shape the Cookies: Using a 1/4 cup measure or ice cream scoop, drop dough onto parchment-lined baking sheets. Flatten slightly and place a small dollop of cheesecake filling in the center. Wrap the dough around to fully enclose the filling.
5. Coat in Cinnamon Sugar: Mix sugar and cinnamon, then roll the cookie balls in the mixture before baking.
6. Bake: Preheat oven to 350°F (175°C). Bake cookies for 10-15 minutes until golden on the bottom and just set; they may look slightly underbaked. Remove from oven and optionally reshape while warm using a round cutter.
7. Final Coating: After cooling enough to handle, dip or brush the front of the cookies again in cinnamon sugar for a crisp coating. Let cool completely on a wire rack; cookies will remain soft inside.
8. Serve: Drizzle caramel sauce over cooled cookies if desired.