Place a baking mat or parchment paper on the baking sheets. Preheat the oven to 400 degrees F.
Whisk the flour, cream of tartar, baking soda, and salt in a mixing bowl. Set aside.
Add the butter, sugar, eggs, and vanilla to a separate mixing bowl. Beat with a mixer until smooth and creamy.
Add the dry ingredients and beat until just blended.
Remove 1/4 of the dough, place it in another small bowl, and set aside. Add several drops of blue coloring to the mixing bowl with 3/4 of the cookie dough. Beat until the dough is a satisfactory shade of blue. Remove the blue dough and place it in another mixing bowl. Remove as much blue dough from the beater(s) as possible.
Add the uncolored dough back to the mixing bowl in which you mixed the blue dough. You do not need to wash the bowl or beater. Add drops of green coloring and beat to mix well. Keep adding green, if necessary, until the desired shade is reached.
Roll blue dough balls with your hands to about 1 to 1 1/2 inches in diameter. Pinch pieces of green dough and stick them randomly onto each blue dough ball. Shapes can mimic continents if desired or be made random because they will change shape when cooked. Once the green shapes are applied, re-roll the balls in your hands to smooth the green pieces.
Place the balls on the prepared cooking sheet about 2 inches apart. Bake for 7-8 minutes or until set. Do not brown. After baking, let the cookies sit for 3 or 4 minutes to set. Then, remove them to a cooking rack.