Preheat the oven to 350°F (175°C). Spread 1/2 cup pecans on a baking sheet and toast for 10-12 minutes. Cool and chop finely.
In a bowl, mix the pecans, flour, baking powder, baking soda, cinnamon, ginger, and salt. In another bowl, whisk oil, sugar, and eggs. Stir in the carrots and fold into the dry ingredients. Pour into an ungreased 9-inch springform pan and bake for 20-25 minutes. Cool completely.
For the cheesecake, beat cream cheese, sour cream, and sugar until smooth. Add eggs one at a time, beating well. Mix in flour, lemon zest and juice, and vanilla.
Pour the cheesecake mixture over the cooled carrot cake. Wrap the pan with foil, place in a roasting pan, and add water halfway up the sides. Bake at 350°F for 1 hour.
Turn off the oven and let the cheesecake sit for 30 minutes. Cool completely and refrigerate for 8 hours or overnight.
For the sour cream topping, whisk together confectioners' sugar, sour cream, vanilla, and salt. Spread over the cheesecake and garnish with chopped pecans. Chill for at least 30 minutes before serving.