These puddings are spicy; if desired, omit the jalapeno, or adjust the amount to taste.
Coat four 3-inch-by-2 1/2-inch ramekins with nonstick cooking spray. Coat with bread crumbs, and tap out excess. Set aside. Heat oven to 450 degrees. Place the carrots, onion, shallots, thyme, rosemary, and jalapeno pepper in a baking pan. Drizzle with olive oil, and toss to coat. Place in the oven to roast, turning over halfway through the cooking, until golden brown and tender, about 25 minutes.
Reduce oven to 375 degrees. Transfer roasted carrots, onion, shallots, and jalapeno pepper to the bowl of a food processor. Process, scraping down the sides as needed, until a thick puree is formed. Transfer puree to a medium bowl. Add the salt, black pepper, pecorino cheese, picked thyme, and chopped rosemary, and stir to combine; set aside.
Using an electric mixer fitted with the whisk attachment, whisk the eggs on high speed until thick, pale, and frothy, and at least tripled in volume, about 5 minutes. Stir 1/4 of the beaten egg into the carrot puree. Fold remaining beaten egg into this mixture. Divide carrot mixture among the prepared ramekins, and place in the oven until puffed and golden brown, about 30 minutes. Remove from oven, and serve.