Preheat oven to 350 degrees F. Lightly grease a 9X13-inch baking pan and set aside. (If using a glass pan, reduce the oven temperature to 325 degrees F.)
For the cake, in a large bowl, whisk together the melted butter, eggs, buttermilk, granulated sugar, light brown sugar, and vanilla until evenly mixed.
Add the shredded carrots and mix until evenly combined.
Add the flour, baking powder, cinnamon, baking soda, salt, and nutmeg, and mix until no dry streaks remain and the batter is evenly combined (try not to over mix in this step).
Spread the batter evenly in the prepared pan.
For the streusel, in a medium bowl, stir together the flour, brown sugar, cinnamon and salt until evenly combined. Add the melted butter and mix well until small clumps form.
Sprinkle the streusel evenly over the cake batter (depending on how long the mixed streusel has been left to sit, it's normal to have to break it into clumpy pieces to sprinkle int his step).
Bake the cake for 40 to 45 minutes until the top springs back lightly to the touch and a toothpick inserted in the center comes out clean or with a few moist crumbs. Add time as needed (the exact time will vary a bit based on differing oven temperatures, pan types, etc.). If the streusel is browning too much but the cake needs more time, place a piece of aluminum foil loosely over the top of the cake while it continues baking. Let the cake cool until room temperature or slightly warm.
For the glaze, in a medium bowl using an electric handheld or stand mixer, mix the cream cheese and butter together until smooth. Add the vanilla and mix.
Add the powdered sugar and mix on low speed, scraping down the sides of the bowl as needed, until the mixture is thick and creamy, 1 to 2 minutes. Add milk to thin the consistency so it is thick but pourable.
Drizzle the glaze over the top of the cooled or just-warm cake OR drizzle over individual servings. The cake is delicious at room temperature or slightly warm.
Notes
Buttermilk: if you don't have storebought buttermilk on hand, whisking together equal parts milk and sour cream is my preferred substitute. Carrots: you'll need about 2 to 3 medium carrots for this recipe. I don't recommend using pre-shredded carrots from the produce section as they are too coarse/thick. I grate the carrots on the small holes of my food processor (no need to peel the carrots first). Lightly pack the shredded carrots into the measuring cup for the amount called for in the recipe.