Preheat your oven to 350°F (175°C). Line two baking sheets with silicone mats if you have them, or use parchment paper for easy cleanup and to prevent sticking.
In a large mixing bowl, add 10 tablespoons of room temperature unsalted butter and beat until light and fluffy. Gradually add in the 3/4 cup granulated sugar and 1/2 cup packed light brown sugar, mixing until smooth. Next, incorporate 1 large egg, 1 cup plain canned pumpkin puree, and 1 teaspoon vanilla extract. Mix all wet ingredients using an electric mixer until well combined.
In a medium bowl, whisk together 2 cups all-purpose flour, 1 teaspoon baking soda, 1/2 teaspoon baking powder, 3/4 teaspoon salt, 1 1/2 teaspoons ground cinnamon, 1/2 teaspoon ground ginger, 1/2 teaspoon ground nutmeg, and 1/8 teaspoon ground cloves. Slowly add these dry ingredients to the wet mixture, mixing until just combined. Don’t overmix—stop as soon as the flour disappears into the batter.
Using a spoon or a 1 1/2 tablespoon cookie scoop (slightly heaping), scoop the dough onto the prepared baking sheets, spacing each about 2 inches apart. Bake one sheet at a time for 13-15 minutes, until the cookies are just browned at the edges. Once removed from the oven, carefully make an indent in the center of each warm cookie with your finger. Leave the cookies on the baking sheet briefly, then transfer to a wire rack to cool completely. I like to make the indent while the cookies are still warm because the dough is softer and less likely to crack.
While the cookies are cooling, make the cheesecake filling. In a medium bowl, blend 5 tablespoons unsalted butter (softened), 4 ounces softened block-style cream cheese, and 1/2 teaspoon vanilla with an electric mixer until very light and fluffy. Gradually add 2 cups confectioners' sugar, mixing until the frosting is smooth. Leave the filling at room temperature for easy spreading. I find the filling is much easier to handle if you don't refrigerate it right away.
Once the cookies have cooled for about 15 minutes, fill each center indent with a generous spoonful of the cheesecake filling from Step 5. Sprinkle a little brown sugar and a pinch of ground cinnamon over the top for added flavor and a beautiful finish. Store the cookies in an airtight container with wax paper between the layers, and keep refrigerated. Bring to room temperature before serving for the best texture and taste. To boost the flavor even more, I like to add a little extra cinnamon on top just before serving.