Step 1: Preheat your oven to 350°F (175°C) and grease an 8x8 inch baking dish with butter.
Step 2: In a medium bowl, whisk together the flour, baking powder, baking soda, cinnamon, ginger, and salt.
Step 3: In a large bowl, beat together 1 cup of brown sugar and 0.5 cup of melted butter until smooth. Add eggs one at a time, followed by the vanilla extract.
Step 4: Gradually stir the dry ingredients into the wet ingredients until just combined. Do not overmix.
Step 5: Fold in the grated carrots and toasted pecans until evenly distributed throughout the batter.
Step 6: Pour the batter into the prepared baking dish and bake for 40-45 minutes, or until a toothpick inserted into the center comes out clean.
Step 7: While the pudding is baking, prepare the brown sugar cream by combining heavy cream, 0.5 cup brown sugar, and 2 tbsp butter in a small saucepan over medium-low heat. Stir until the sugar is dissolved and the sauce is slightly thickened (about 5-8 minutes).
Step 8: Once the pudding is removed from the oven, poke holes all over the surface and pour half of the warm brown sugar cream over it. Let it soak for 10 minutes before serving with the remaining sauce.