Preheat oven to 375°F (190°C). Line baking sheets with parchment paper or a silicone baking mat; set aside.

If using freshly made browned butter, transfer 1 cup browned butter to a heatproof bowl and allow it to cool slightly until warm but not hot.

In a large bowl, combine 1 cup browned butter, 1 cup white sugar, and 1/2 cup brown sugar. Beat or stir until the sugars are well incorporated and the mixture is uniform, scraping down the sides of the bowl as needed.

Add 2 eggs and 2 teaspoons vanilla extract to the butter-sugar mixture. Beat until the mixture is pale and thick.

In a separate bowl, whisk together 2 3/4 cups all-purpose flour, 2 teaspoons cream of tartar, 1 teaspoon baking soda, and 1/2 teaspoon salt.

Add the dry ingredients to the wet ingredients and mix just until combined; avoid overmixing.

Cover the dough and chill for 45 minutes or refrigerate overnight.

In a small bowl, mix 4 tablespoons white sugar and 1 tablespoon ground cinnamon for rolling.

Scoop or measure dough into balls about 1.5 tablespoons each. Roll each ball in the cinnamon-sugar mixture to coat.

Place the coated dough balls 2 inches apart on the prepared baking sheets.

Bake 8 to 10 minutes, until the edges are set and golden and the centers are puffed.

Let cookies cool on the baking sheet for about 5 minutes, then transfer to a wire rack to cool completely.