Melt small amount of sugar:Sprinkle 1/3 cup sugar in a heavy, 3 1/2-qt. Dutch oven; cook over medium heat, stirring occasionally, 3 minutes or until sugar is melted and syrup is a very light golden brown. Remove from heat.
Boil sugar and milk:Stir together 1 Tbsp. flour and 2 1/2 cups sugar in a large saucepan; add milk, and bring to a boil over medium-high heat, stirring constantly. Remove from heat.
Warm up caramel:Set Dutch oven with caramel over low heat until sugar is liquefied if cooled. Stir until smooth.
Add hot milk mixture to caramelized sugar:Gradually pour about one-fourth hot milk mixture into caramelized sugar, stirring constantly until smooth. Take care: the mixture will bubble and sputter as the liquid is absorbed into the caramel. Gradually stir in remaining hot milk mixture until smooth.
Cook caramel frosting:Cover, and cook over low heat 2 minutes. Increase heat to medium; uncover and cook, without stirring, until a candy thermometer reaches 238°F (soft-ball stage, about 10 minutes). Add butter, stirring until blended. Remove from heat and let stand, without stirring, until temperature drops to 110°F (about 1 hour).
Mix frosting in stand mixer:Pour caramel into bowl of heavy-duty electric stand mixer. Add vanilla, and beat at medium speed (setting 4) with whisk attachment until spreading consistency (about 20 minutes).
