Preheat the oven to 350°F (175°C)
Prepare a 9-inch springform pan by greasing it well and wrapping the base with parchment paper to prevent water from seeping in.
In a large mixing bowl, sift the flour, baking soda, cinnamon, nutmeg, salt, and 0.5 teaspoon of the sugar (cake portion). Mix to combine.
In another bowl, whisk the vegetable oil, 1.5 cups of sugar (cake portion), 4 large eggs, and 2 teaspoons of vanilla extract until well combined.
Fold in the grated carrots and chopped pecans (if using), then gradually mix in the dry ingredients until the batter is smooth.
Pour the batter into the prepared pan and bake for 25–30 minutes, or until the top is firm and a toothpick inserted in the center comes out clean.
While the cake is cooling, prepare the cheesecake. In a separate bowl, beat the cream cheese (cheesecake portion), 1.5 cups sugar (cheesecake portion), 3 eggs, 1 teaspoon vanilla extract, and 1 cup sour cream until smooth and creamy.
Pour the cheesecake mixture directly over the partially baked and cooled carrot cake base. Return to the oven and bake for an additional 20–25 minutes, or until the edges are firm but the center still slightly jiggles. Let the cheesecake cool completely before chilling in the refrigerator for at least 4 hours or overnight.
To make the frosting: Beat the cream cheese (frosting portion) with 2 teaspoons of vanilla extract until smooth. Spread the frosting over the chilled cheesecake before serving.