Preheat oven to 350°F. Line 2 baking sheets with silicone baking mats or parchment paper.

In a large bowl with an electric mixer, cream together the butter, sugars, and yogurt for 1 full minute. Beat in the eggs, one at a time, until incorporated. Add vanilla, baking soda, salt, and cream of tartar and mix on low until combined. Stir in flour until just combined, then gently stir in chocolate chips.

Drop the dough by large spoonfuls onto the prepared baking sheets (I used a #30 scoop, which is just over 2 tablespoons per scoop). Place 12 on each baking sheet, leaving space between each scoop of dough.

Bake for 18-20 minutes in the preheated oven, or until edges are nicely browned. Let cool on the baking sheets for a minute or two before transferring to wire racks to cool completely. Repeat with remaining dough.

Once the cookies are completely cool, make the icing. In a medium bowl, combine butter, powdered sugar, vanilla, and yogurt. Stir with a fork until smooth. If it's too thick, add a little more yogurt. If it's too thin, add a little more powdered sugar. You want it to be thick enough to hold its shape if you plan to write with it.