Break off a quarter of the cookie dough (about 6 tablespoons) and divide into 2 pieces. Put each piece in a separate small bowl. Tint one piece with 4 drops of purple gel food coloring. Tint the other piece with 3 drops of purple gel food coloring and 2 drops of white food coloring.
Roll out the remaining un-tinted cookie dough between 2 pieces of parchment to an 1/8-inch-thick round (about 12 inches in diameter). Remove the top piece of parchment and reserve.
Break off a chickpea-size piece of dark purple dough and roll into a rough ball. Place on the rolled-out dough. Arrange 6 roughly rolled chickpea-size pieces of the light purple dough around the dark purple dough in a flower pattern (these will become the petals) about 1 1/2-inches round. Make more flowers with the remaining tinted dough: leave about a 1/2 inch between each flower shape and alternate between dark purple and light purple dough in the center and petals.
Top the dough with the parchment and gently roll out until the dough is flat and even. Slide the round onto a rimmed baking sheet and freeze until firm, about 1 hour.
Use a 1 3/4-inch 6-petal flower cookie cutter to cut out each flower shape. Arrange on 2 parchment-lined baking sheets, spacing the cookies 1 inch apart. Freeze until firm, about 30 minutes.
Preheat the oven to 350 degrees F. Bake the cookies until just barely golden on the edges, 10 to 12 minutes. Let the cookies cool on the baking sheet on a rack.
Pipe a small yellow dot of icing in the center of each cookie. Let sit for 15 minutes to set.
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