In a large bowl, using a mixer set on medium-high speed, beat the butter until fluffy. On medium speed, beat in the sugar, baking powder and salt. Beat in the egg and vanilla. On medium-low, gradually beat in the flour. Shape the dough into a disk; wrap in plastic and chill for 2 hours.
Divide the dough into 3 sections. Return 2 sections to the refrigerator. Knead the red and brown food colorings into the remaining dough until a reddish-brown color is reached; rewrap the dough and chill 30 minutes.
Grease 2 baking sheets. Divide the untinted dough into 18 equal pieces. Using your hands, roll each piece into a 3-inch bun. Place the buns onto the prepared baking sheets. Using the handle of a wooden spoon, make a deep indentation in the center of each bun.
Divide the tinted dough into 18 equal pieces. Roll each piece into a 3-inch log; round the ends to resemble hot dogs. Place in center of split dough buns on baking sheets; freeze assembled hot dogs for 30 minutes.
Preheat the oven to 350F. Bake the cookies until set, 12 to 14 minutes. Transfer the cookies to wire racks and cool completely.
Using a tube of yellow icing fitted with a round tip, pipe a squiggly line onto each cookie to resemble mustard; sprinkle green sugar on top to resemble relish.
Tip: Don't Have a Round Tip?
If you do not have a round tip designed to fit onto a tube of decorating icing, spoon the icing into a small resealable plastic bag and snip off 1 corner.