Start by peeling and finely dicing the Gala apples, then toss them in a mixing bowl with dark brown sugar, ground cinnamon, salt, and lemon juice. In a medium skillet over low-medium heat, cook the apple mixture for about 10 minutes, stirring occasionally, until the apples are soft and a rich caramel color forms. Once done, transfer them to a bowl and chill in the refrigerator.

In a medium saucepan, melt the European-style unsalted butter over medium heat, stirring frequently. Continue to cook until the butter turns a deep golden brown and emits a nutty aroma, which typically takes about 5-7 minutes. Remove the pan from heat, let it cool for 10 minutes, placing it in the fridge to speed up the process while you prepare the other ingredients.

In a separate bowl, whisk together all-purpose flour, baking soda, cream of tartar, ground cinnamon, salt, and cornstarch. Ensure there are no lumps. Setting it aside allows the ingredients to meld.

In a large mixing bowl, combine the cooled brown butter with dark brown sugar, granulated sugar, and vanilla bean paste or extract. Mix well until smooth. Next, add the egg and egg yolk while continuing to beat until fully incorporated. Gently fold in the previously cooled caramelized apple mixture.

Gradually add the dry ingredient mixture to the wet ingredients, gently stirring until just combined. Avoid over-mixing; cover the dough with plastic wrap and chill in the refrigerator for at least 2-3 hours.

Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper. In a small bowl, mix granulated sugar and additional ground cinnamon for rolling the cookies.

Using a cookie scoop, portion out the chilled dough and roll each ball in the cinnamon sugar mixture to coat thoroughly. Place the rolled balls on the lined baking sheet, keeping them about 2 inches apart.

Bake the Caramelized Apple Snickerdoodles in the preheated oven for 12-14 minutes or until the edges are set and have a slight golden hue. Allow the cookies to cool on the baking sheet for a few minutes before transferring them to a wire rack.