Preheat the oven to 375°F. Line a baking sheet with parchment paper and set aside.
Brown the butter: Cut the butter into cubes and melt it in a saucepan over medium heat, stirring often. It will melt, then foam, and the milk solids will turn golden brown. Watch closely toward the end so it doesn’t burn. Remove from the heat and cool for 5–10 minutes, until warm but not hot to the touch. Pour into a large mixing bowl.
Add the sugar to the cooled brown butter and beat on high speed until smooth, about 3 minutes.
Beat in the sourdough discard, egg, egg yolk, and vanilla until combined. Scrape down the bowl as needed.
Set a sifter over the bowl and sift in the flour, cream of tartar, baking soda, and salt. Mix until the dough comes together, pausing once or twice to scrape the bowl. The dough should be soft, not sticky. If it feels sticky, mix in 1–2 tablespoons more flour.
Make the topping: Combine the sugar and cinnamon in a small bowl.
Scoop the dough into balls (about 2 tablespoons each) and roll them in the cinnamon sugar. Place them on a plate and chill for 30 minutes.
After chilling, space the dough balls 3 inches apart on a parchment-lined baking sheet. Bake for 9 to 12 minutes, until the edges just look set and dry. The centers may look slightly underbaked. Don’t overbake if you want that soft texture.
Let the cookies cool on the pan for 10 minutes, then transfer to a wire rack to cool completely.
Notes
Don’t skip the chill: A 30-minute rest keeps the cookies thick and chewy, and helps prevent overspreading.Watch your butter: Brown it just until golden and nutty. If it gets too dark, and it’ll taste bitter.Bake until just set: The edges should look dry, but the centers should still look soft. They’ll finish setting as they cool.Don’t skip the cream of tartar: It’s what gives snickerdoodles their signature tang and crackly texture