Step 1: Preheat your oven to 350°F (175°C) and grease a 9-inch square baking dish with butter.
Step 2: In a large bowl, cream together the softened butter and 1 cup of dark brown sugar until light and fluffy.
Step 3: Beat in the eggs one at a time, followed by the vanilla extract and the finely grated carrots.
Step 4: In a separate bowl, whisk together the flour, baking powder, cinnamon, ginger, and salt.
Step 5: Gradually add the dry ingredients to the wet ingredients, alternating with the milk, mixing until just combined.
Step 6: Fold in the toasted chopped pecans gently using a spatula.
Step 7: Pour the batter into the prepared baking dish and bake for 40-45 minutes, or until a skewer inserted into the center comes out clean.
Step 8: While the pudding bakes, prepare the brown sugar cream by combining the heavy cream, brown sugar, and butter in a small saucepan over medium heat.
Step 9: Bring the sauce to a gentle simmer for 5 minutes until thickened and glossy, then remove from heat.
Step 10: Poke holes in the warm pudding, pour half the sauce over it, and serve the rest on the side.