Beat the butter and sugar until light and fluffy, about 2–3 minutes. Beat in the egg and vanilla. Add the flour and salt and mix just until a soft dough forms. Cover and refrigerate for 30 minutes.
Preheat the oven to 350°F(175°C) and line a baking sheet with parchment paper. Scoop the dough into 1 1/2-tablespoon portions and roll into balls. Place them 2 inches apart, flatten slightly, then press your thumb or the back of a teaspoon into the center to make a well.
Bake for 10–12 minutes, until the edges are lightly golden. If the centers puff up, gently press the wells down again while the cookies are still warm. Transfer to a wire rack and cool completely.
Beat the cream cheese until smooth. Add the pumpkin, brown sugar, pumpkin pie spice, cinnamon, heavy cream, and vanilla. Beat until smooth and creamy. Refrigerate for at least 30 minutes.
Spoon or pipe the pumpkin filling into the cooled cookies. Top with whipped cream and a sprinkle of cinnamon or nutmeg, if desired.