For the carrot cake: Adjust an oven rack to the middle position and preheat to 350 degrees F. Spread 1/2 cup of the pecans on a rimmed baking sheet and bake until golden and toasted, 10 to 12 minutes. Allow to cool, and then finely chop.
Combine the pecans, flour, baking powder, baking soda, cinnamon, ginger and 1/4 teaspoon salt in a large bowl. Whisk together the oil, sugar and eggs in a separate bowl. Stir the carrots into the egg mixture. Fold the carrot-egg mixture into the flour mixture until just combined. Pour into an ungreased 9-inch springform cake pan and tap it on the counter to even out the batter. Bake until the cake bounces back when pressed and a toothpick inserted in the middle comes out clean, 20 to 25 minutes. Cool completely. The cake can be made a day ahead and stored in the pan.
For the cheesecake: Preheat the oven to 350 degrees F. Beat the cream cheese, sour cream and granulated sugar in a stand mixer fitted with a paddle attachment until smooth and creamy, about 5 minutes, scraping down the sides of the bowl as needed. Add the eggs one at a time, beating well after each addition. Beat in the flour, lemon zest and juice and vanilla, about 1 minute.
Pour the cheesecake mixture over the carrot cake. Wrap the bottom and sides of the pan with a large piece of foil. Put in a roasting pan or a large baking dish and fill halfway up the sides of the cake pan with water. Bake until the cheesecake is pale yellow and just jiggles slightly in the center, about 1 hour. Turn off the oven and let the cheesecake rest in the hot oven for 30 minutes. Remove, run a sharp knife around the edge and let cool completely on a rack. Cover and refrigerate 8 hours or overnight.
For the sour cream topping: Whisk together the confectioners' sugar, sour cream, vanilla and salt. Spread on top of the cheesecake; garnish with chopped pecans. Refrigerate for at least 30 minutes before serving. Cut into slices and serve.
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