Freeze 1 stick (8 tablespoons) of butter until completely frozen.
Move the oven rack to the lower third of the oven.
Preheat the oven to 400 degrees F. and line a tray with parchment paper or a nonstick liner and set aside.
In a large bowl, toss together the flour, 1/4 cup white sugar, brown sugar, baking soda, baking powder, and salt.
Remove the frozen butter from the freezer and grate it on the large side of a cheese grater.
Working quickly cut the frozen butter pieces into the flour mixture with a pastry blender or two butter knives until the butter is worked into the flour.
Toss in the miniature chocolate chips.
In another bowl, mix together the vanilla extract, Greek yogurt, and egg. Mix until well combined.
Add the wet to dry and mix until just combined with a fork. Use your hands to press the dough into a large ball. The dough will be slightly sticky and it may also seem dry but it comes together as you press it into a ball. The less you mix the better the texture will be. Over-working/ mixing the dough yields a denser end product.
On a floured surface, form the mixture into a 7-8 inch circle that is about 3/4th an inch thick.
Sprinkle with the remaining 1 teaspoon of sugar.
Cut the dough into 8 triangles and then place the triangles on a cookie sheet about 1 inch apart.
Bake until lightly golden about 14-16 minutes.
Remove, allow to slightly cool and then top with the glaze.
For the glaze, whisk the milk or heavy cream, powdered sugar, and vanilla extract together until smooth. Add more milk/heavy cream as needed, but you do want the glaze quite thick.