In a medium bowl, whisk together the dry ingredients: flour, baking powder and salt.
Beat together the butter, sugar and confectioners' sugar in a large bowl with an electric mixer on medium-high, until light and fluffy (about 2 minutes).
Add in the egg yolks and vanilla and mix until smooth. Reduce mixer speed to medium-low and slowly add in the dry ingredients. Mix the dough until it just comes together.
Divide the dough into 6 pieces: 2 large pieces (about 7 oz. each), 2 medium pieces (about 5 oz. each) and 2 small pieces (about 3 oz. each).
Tint 1 large piece pink or red and the other large piece orange - work the food coloring into he dough until it's all one color. Next, tint 1 medium piece yellow and 1 medium piece green. Lastly, tint one small piece blue and one purple.
Using gloves hands, roll the purple dough into a 7" log (about 3/4" thick) on a piece of parchment paper dusted with flour. Using a rolling pin, roll out the remaining colors of dough into rectangles on separate sheets of floured parchment paper.
Use these dimensions for each color: Blue - 3.5" x 7", Green - 5" x 7.5", Yellow - 6"x8", Orange - 7"x8.5", Red - 8"x9"
Tightly wrap the blue dough around the purple log, keeping the dough in a log shape; pinch the seam to close. Continue to wrap with the remaining colors: green, then yellow, orange and pink/red.
Wrap the whole log in plastic wrap and freeze until very firm, about 1 hour.
Preheat the oven to 375F. Time the ends of the dough log, then slide into 3/8" thick rounds and slice each round in half.
Arrange the cookies on 2 parchment lined baking sheets, 1" apart. And refrigerate until ready to bake, at least 15 minutes.
Bake 1 pan at a time, until the cookies are just set around the edges and golden on the bottom 10-15 minutes. Allow to cool for 5 minutes on the pan before transferring to a wire rack to cool completely.
Once the cookies have cooled, melt the white chocolate (microwave for 30 second increments stirring in between, until smooth) and transfer to a squeeze bottle or Ziploc plastic bag with the corner snipped
Pipe cloud shapes with the white chocolate or dollop dots to stick the mini marshmallows to the cookies.