For cookies, place butter into a light-colored saucepan over medium-low heat. Cook, stirring constantly, until amber in color, 7 to 10 minutes. Immediately pour browned butter into a large heat-safe mixing bowl, and let cool for 20 minutes.

Once browned butter has cooled, add cream cheese, 1/2 cup white sugar, brown sugar, corn syrup, salt, and nutmeg. Beat mixture with an electric mixer until smooth and combined, about 3 minutes. Add egg and vanilla, and beat for an additional 1 minute. Beat in flour, cornstarch, and baking soda until just combined. Cover bowl and place in the freezer until the dough is easier to handle, about 20 minutes.

Line a baking sheet with parchment paper. Pour 1/3 cup sugar into a small bowl.

Remove dough from freezer. Scoop out 1 1/2 tablespoons of the dough and drop into the sugar, coating it. Roll dough into a ball, then roll again in sugar until thoroughly coated. Place dough ball onto the parchment-lined baking sheet. Using your thumb, or the handle of a wooden spoon or spatula, make an indentation into the dough about 1 inch wide and slightly less than a 1/2 inch deep. Repeat this process with remaining cookie dough.

Freeze formed dough until chilled, about 15 minutes, or refrigerate for about 30 minutes.

Preheat the oven to 350 degrees F (180 degrees C). Line another baking sheet with parchment paper.

For filling, whisk sweet potato puree, sugar, egg, half-and-half, lemon juice, vanilla, cinnamon, ginger, nutmeg, and salt until smooth and combined.

Remove 9 chilled cookies from the freezer and space them 2 inches apart on the prepared baking sheet. Fill each cookie with 2 teaspoons filling.

Bake cookies in the preheated oven until edges of cookies begin to turn golden brown, 12 to 15 minutes. Cool cookies on baking sheet for 5 minutes, then carefully remove to a wire rack to cool completely.

Repeat process of filling and baking cookies, 1 baking sheet at a time, until all cookies are baked.