In a large bowl, mix together the grated carrots, graham cracker crumbs, brown sugar, and melted butter. Press this mixture into the bottom of a 9-inch springform pan. Bake for 10-12 minutes, or until lightly browned.
In a blender or food processor, puree the strawberries and balsamic vinegar. Pour the mixture into a saucepan and cook over medium heat until it reduces to a syrupy consistency.
In a large mixing bowl, beat the cream cheese with an electric mixer on medium speed until smooth. Add the white sugar, eggs, and vanilla extract, and continue to beat until well combined.
Spread the strawberry-balsamic mixture over the carrot cake base in the springform pan, then pour the cream cheese mixture over the top.
Bake for 40-45 minutes, or until the cheesecake is set in the center. Refrigerate the cheesecake for at least an hour.
Melt the chocolate chips and cream together in a double boiler or microwave, stirring until smooth. Spread this mixture over the chilled cheesecake.
Sprinkle the chopped pecans over the chocolate layer. Allow the chocolate to cool and harden before slicing and serving.