Pulse radishes in a food processor until finely minced, and transfer to a medium bowl. Stir in goat cheese, chives, chervil, and 1 teaspoon salt. Season with pepper. Cover, and refrigerate for 30 minutes.
Meanwhile, prepare an ice-water bath. Bring a medium pot of salted water to a boil. Using a vegetable peeler, shave carrots lengthwise into thin ribbons. (You will need 36 ribbons, each 8 inches long.) Boil carrot ribbons until softened, about 2 minutes. Transfer to ice-water bath. Drain, then pat dry with paper towels.
Roll each carrot ribbon, tapering 1 end, to 1/2 inch in diameter. Transfer carrot rolls to a parchment-lined baking sheet with narrow end up.
Transfer chilled radish mixture to a pastry bag fitted with a 1/4-inch plain tip. Pipe about 1/2 teaspoon radish mixture into center of each carrot roll. Refrigerate until firm, at least 30 minutes (or up to 1 hour). Garnish with chervil sprigs, and serve immediately.