01 - Combine cranberries, sugar, orange juice, and zest in a small saucepan. Cook over medium heat, stirring occasionally, until cranberries burst and mixture thickens, about 7 minutes. Allow to cool completely, then puree until smooth.
02 - Whisk together flour, baking powder, and salt in a medium bowl. In a separate large bowl, beat butter and sugar until light and fluffy, approximately 2 to 3 minutes. Incorporate egg and vanilla extract. Gradually add dry ingredients, mixing until just combined.
03 - Divide dough into two equal portions. Roll each portion between parchment sheets into a 9 by 8-inch rectangle. Spread half of the cranberry mixture evenly over each rectangle, leaving a 1/2-inch border. Carefully roll each into a tight log, wrap in plastic, and chill for at least 1 hour until firm.
04 - Preheat oven to 350°F. Line baking sheets with parchment paper. Slice chilled logs into 1/2-inch rounds and place 2 inches apart on prepared sheets. Bake for 10 to 12 minutes or until edges turn lightly golden. Transfer to wire racks to cool completely.
05 - Beat powdered sugar, softened butter, milk, vanilla extract, and salt until smooth and spreadable. Once cookies are fully cooled, spread a thin layer of frosting on each cookie.