Grate your carrots using the small holes of a box grater, then wrap in a clean kitchen towel and squeeze firmly over the sink until no more liquid drips out. You should have approximately 95 grams of carrot after squeezing.
In a large bowl, combine the water, sourdough starter, bread flour, salt, brown sugar, vanilla extract, cinnamon, nutmeg, ginger, cloves, orange zest, and squeezed carrot. Mix everything together at once until no dry flour remains. The dough will look sticky and shaggy, which is completely normal.
Cover the bowl and let the dough rest at room temperature for 1 hour.
After resting, perform one round of stretch and folds to build strength in the dough. Wet your hands, grab one edge of the dough, stretch it up, and fold it over itself. Rotate the bowl 90 degrees and repeat 4 to 6 times total until you have worked all the way around the dough. Slap and fold the seam onto the bottom of the bowl.
Cover and bulk ferment at room temperature for 8 to 12 hours, or until the dough has doubled or tripled in size, looks full of bubbles, and feels jiggly when you gently shake the bowl. Watch the dough, not the clock, as timing varies with kitchen temperature.
When ready to shape, line a 9x13 inch pan with parchment paper. Melt the butter and stir in the vanilla extract, then pour into one shallow bowl. In a second bowl, whisk together the brown sugar, cinnamon, nutmeg, ginger, and cornstarch until evenly combined.
Oil your hands lightly. Grab a golf ball sized hunk of dough, dunk it fully in the vanilla butter letting excess drip off, then roll it generously in the spiced coating until completely covered. Place it in the prepared pan. Repeat until the pan is full with all balls touching.
Cover the pan loosely and let proof for 1 to 2 hours at room temperature until the balls are visibly puffy, jiggly, and have grown to almost doubled.
Preheat your oven to 425 degrees F (218 C)
Once proofed, use an oiled finger to press one firm dimple into the center of each ball, pushing the coating down into the middle of the dough. One press per ball is all you need.
Bake for 20 to 25 minutes until the tops are deep golden brown and the coating has caramelized.
Remove from the oven and let cool in the pan for 5 minutes.
While the focaccia cools, make your drizzle. For the cream cheese drizzle, beat the softened cream cheese and milk together until smooth and lump free, then add the powdered sugar and vanilla and whip until glossy and pourable. For the vanilla glaze, simply whisk together the powdered sugar, cream or milk, and vanilla until smooth.
Drizzle generously over the warm focaccia, letting it fall into the gaps between the balls. Serve warm.