Heat 2 tablespoons butter with the olive oil in a large pot over medium heat until foaming. Add the leeks, ginger, a pinch of salt and a few grinds of black pepper and cook, stirring occasionally, until softened but not browned, 10 to 12 minutes.
Stir in the carrots. Add the milk, nutmeg, 2 more tablespoons butter, 1/4 teaspoon salt and 1/2 cup water. Increase the heat and bring to a simmer. Reduce the heat to medium low, cover and cook, stirring occasionally, until the carrots are very tender, about 30 minutes. Transfer the carrot mixture to a blender and puree until smooth.
Add the remaining 4 tablespoons butter to the blender a little at a time and puree until the butter melts and the puree is creamy; season with salt and the cayenne. Transfer to a bowl and top with more nutmeg.
Make the mash through the end of step 2 up to 1 day ahead; cover and refrigerate. Gently reheat, then complete step 3.