Make the crust: Preheat oven to 350 degrees. Pulse pecans in a food processor until finely ground. Place in a medium bowl with graham crackers, and stir in butter, sugar, and salt.
Wrap outside and bottom of a 9-inch springform pan in 2 layers of foil. Press crumb mixture firmly into bottom of pan. Bake until golden brown around edges and firm, about 15 minutes. Let cool on wire rack. Reduce oven temperature to 325 degrees.
Make the cheesecake: Melt butter in a medium saute pan over medium heat. Add carrots and 1/2 cup sugar, and cook, stirring often, until carrots are soft, 3 to 4 minutes. Transfer with pan juices to a bowl, and let cool.
Beat cream cheese and remaining cup sugar with a mixer on medium speed until fluffy, about 4 minutes. Add eggs, 1 at a time, scraping down sides of bowl as needed. Add carrots and pan juices, nutmeg, cinnamon, ginger, salt, and vanilla, and beat until incorporated. Pour into crust.
Set springform pan in a large roasting pan. Fill roasting pan with enough boiling water to reach halfway up the sides of springform pan. Bake cheesecake until set but slightly wobbly in center, 80 to 90 minutes.
Transfer springform pan to a rack, and remove foil. Let cool. Refrigerate, uncovered, at least 6 hours (or overnight; cheesecake will keep, covered and refrigerated, for up to 3 days). Run a hot knife around edges of cheesecake to loosen, then remove sides of pan. Decorate with marzipan carrots just before serving.