Cream the base — Beat the room temperature vegan butter and brown sugar together until the mixture looks like a pale, fluffy paste. You aren't looking for a total dissolve, but it should feel cohesive and smell like caramel.
Incorporate the aromatics — Stir in the vanilla extract and the finely shredded carrots. The mixture might look slightly broken or wet at this stage because of the carrot juice, but don't worry; the flour will fix it.
Whisk the dry goods — In a separate bowl, combine the flour, cinnamon, nutmeg, ginger, baking soda, and salt. Whisking them beforehand prevents a "spice clump" where one lucky person gets a mouthful of pure ginger.
Combine the dough — Slowly fold the dry ingredients into the wet. If the dough feels too stiff to scoop, add the dairy-free milk one teaspoon at a time until it holds together.
Bake to set — Scoop rounded tablespoons onto a tray and bake at 350°F for about 10–12 minutes. Look for the edges to turn a golden brown while the center still looks slightly underdone and soft.
Whip the frosting — While the cookies cool completely, beat the vegan butter, cream cheese, and powdered sugar until smooth. Only frost once the cookies are stone cold, or you'll end up with a glazed mess.