In a large bowl, beat the sugar and butter together until the butter is softened and creamy. Add in the eggs, milk, and vanilla, and continue mixing for about 30-45 seconds. Gradually add in the flour, and mix until all of the ingredients are combined into the dough.
Divide the dough in half and place in separate bowls. Add red to one bowl and green to the other bowl, mixing the coloring into each portion of the dough.
Cover and chill the dough in the freezer for about 10 minutes. Once you remove the dough from the freezer, place it on a lightly floured surface and roll out each color of dough into a rectangle shape. (If necessary, place back in the refrigerator or freezer for another 10 minutes. You want to make sure the dough is firm enough to roll) Brush the top of both rectangles with some of the egg white mixture before stacking on top of each other. Trim the edges of the rectangle so the edges of the dough are straight. Beginning with the longer end of the rectangle, gently roll the dough and pinch the seam. Wrap and place in the freezer for about 10-15 minutes.
Preheat the oven to 350°F/180°C, and line 2 cookie sheets with parchment paper.
Once the dough is chilled again, slice into small discs about ¼ inch in thickness. Place onto a parchment paper-lined baking sheet, leaving room for them to spread during baking. Bake about 13-15 minutes, until the cookies are set without turning golden brown.
Remove from the oven and allow cookies to cool.