Preheat oven to 350° (180°C). Spray a 9-inch round cake pan with baking spray with flour.
For cake: In a medium bowl, whisk together flour, baking powder, salt, cinnamon, and ground ginger.
In a large bowl, whisk together brown sugar, oil, applesauce, eggs, and vanilla. Whisk in flour mixture just until dry ingredients are moistened; stir in carrots, pecans, and crystallized ginger. Spread batter into prepared pan.
Bake until a wooden pick inserted in center comes out clean, 45 to 50 minutes. Let cool in pan on a wire rack for 15 minutes. Gently remove from pan, and let cool completely on a wire rack.
For topping: In a medium heavy-bottomed saucepan, bring brown sugar, cream, butter, and salt to a boil over medium-high heat, stirring until sugar dissolves. Cook, stirring constantly, until an instant-read thermometer registers 225°F to 230°F (107°C to 110°C), 1 to 2 minutes. Remove from heat. Gradually whisk in confectioners’ sugar until smooth. Pour onto cooled cake, gently spreading smooth; sprinkle with pecans. Let stand for 15 minutes before slicing. Store in an airtight container for up to 3 days.