Heat oven to 350ºF. Line a cupcake pan with paper liners and set aside.
For the cupcakes: In a bowl sift together flour, baking powder, baking soda and salt; set aside.
In an electric mixer fitted with a paddle attachment, beat butter and sugar together until light and fluffy, about 5 minutes. Add eggs, one at a time, beating between each addition. Scrape down sides of the bowl; add vanilla and stir to combine.
Add flour mixture alternately with the warm water in three additions, beating between each (beginning and ending with flour mixture).
Divide batter evenly among liners, about three-quarters full each. Bake until tops spring back when touched, 15 to 17 minutes. Let cool completely.
For the Caramel Icing: Combine ⅓ cup shortening, ¼ cup hot milk, 2 ½ cups confectioners' sugar, and vanilla and beat until smooth, about 2 min.
Boil together 1 tbsp shortening, ½ cup cold milk, corn syrup and brown sugar until temperature reaches 235°F (soft-ball stage) on a candy thermometer, about 10 min. Pour over shortening mixture and beat until smooth and creamy, about 2 min. Add ½ cup confectioners' sugar to create a smooth and creamy frosting. If icing isn't smooth enough, add 1 tbsp hot water.
Transfer icing to a large piping bag fitted with a Wilton 1M open star tip and pipe rosettes on top of each cupcake.
Make and cool the cupcakes before making the icing. Pipe the icing immediately after preparing.