Preheat your oven to 350°F (175°C), and line a baking sheet with parchment paper.
In a medium saucepan over medium heat, melt the butter, whisking constantly. It will foam before turning golden brown, with brown specks forming at the bottom. Once the butter is browned, immediately transfer it to the bowl of a stand mixer fitted with the paddle attachment. Let it cool for about 5 minutes.
Add the brown sugar and granulated sugar to the browned butter and mix until well combined. Then, mix in the egg, egg yolk, Greek yogurt, and vanilla extract until smooth and creamy.
Gradually add the dry ingredients (flour, baking soda, cream of tartar, cinnamon, and salt) to the wet mixture, mixing until just combined.
Use a cookie scoop or spoon to portion the dough onto the prepared baking sheet, spacing the cookies about 2 inches apart. Bake for 8-10 minutes, or until the edges are golden brown and the centers look slightly underbaked
Remove the cookies from the oven and immediately sprinkle them with the remaining 1/3 cup of toffee pieces. Let them cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.