Preheat oven to 350°F (175°C) and line a 9-inch square pan with parchment paper.
Combine graham cracker crumbs, melted butter, and 1/3 cup sugar in a bowl until it resembles wet sand.
Press the crumb mixture firmly into the bottom of the prepared pan, slightly up the sides.
Bake the crust for 10 minutes, then remove from the oven and set aside to cool slightly.
In a large bowl, beat the softened cream cheese with 1/2 cup sugar and vanilla until smooth.
Add the eggs one at a time, mixing on low speed until just combined.
Remove about 1/2 cup of the plain cheesecake batter and spread it evenly over the crust.
Stir the pumpkin purée, cinnamon, cloves, and nutmeg into the remaining cheesecake batter until smooth.
Pour the pumpkin cheesecake mixture over the plain layer and spread evenly.
Bake for 30 minutes, or until the center is set but slightly wobbly.
Cool at room temperature, then refrigerate for at least 3 hours until completely firm before cutting into squares.