Place the 1 cup of unsalted butter in a small saucepan over medium-low heat and allow it to melt completely.
Once liquefied, reduce the heat to a gentle simmer. Continue cooking for 10–15 minutes, stirring from time to time, until the butter takes on a golden-brown color, develops browned milk solids, and smells distinctly nutty and toasted.
Immediately pour the browned butter into a mixing bowl and leave it to cool while preparing the cider caramel.
Using the same saucepan, pour in the 2 cups of apple cider and bring it to a boil over medium-high heat.
Continue boiling for approximately 15–20 minutes, or until the cider has concentrated to roughly 1/3 cup.
Turn the heat down to medium-low. Whisk in the 1 cup of cane sugar until it has completely dissolved into the reduced cider.
Add the 4 tablespoons of salted butter followed by the 1/2 cup of heavy cream. Stir continuously and maintain a low simmer for about 5 minutes, allowing the mixture to thicken.
Remove the saucepan from the heat. Stir in the 1/2 teaspoon cinnamon, 1/4 teaspoon sea salt, and 1/2 teaspoon vanilla extract.
Let the caramel cool slightly, then pour it into a glass jar and reserve it for glazing the finished cookies.
Add the 3/4 cup cane sugar and 1 cup packed brown sugar to the bowl containing the cooled browned butter.
Using either a stand mixer fitted with the paddle attachment or a hand mixer, beat the ingredients on medium-high speed until thoroughly blended. Because the browned butter remains liquid rather than creamed, the mixture is not expected to become light and fluffy.
Add the 3/4 cup sourdough starter and the 3 egg yolks. Mix on medium speed until evenly incorporated.
Blend in the 1/2 cup apple butter and 1 1/2 teaspoons vanilla extract, continuing only until the mixture becomes smooth and uniform.
To the wet mixture, add the 3 cups flour, 1 teaspoon baking soda, 1 teaspoon baking powder, 1 teaspoon sea salt, 1 teaspoon cinnamon, 1/2 teaspoon ginger, 1/4 teaspoon allspice, 1/4 teaspoon cardamom, and 1/4 teaspoon nutmeg.
Mix on low speed just until the dry ingredients disappear into the dough, stopping to scrape down the bowl when necessary. Avoid excessive mixing once the flour has been added.
Add the 3/4 cup dark or bittersweet chocolate chips, 1 cup chopped pecans, and 1 1/2 cups chopped dried apples. Mix only until the additions are evenly dispersed throughout the dough.
Cover the bowl securely with plastic wrap and refrigerate the cookie dough for at least 4 hours. This resting period allows the dough to become firm while the flavors develop.
For a more pronounced sourdough character and a longer fermentation period, the covered dough may remain refrigerated for 24–72 hours.
Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper.
Using a large cookie scoop, divide the chilled dough between the prepared baking sheets, leaving approximately 2 inches of space between each portion.
For consistently sized cookies, portion approximately 68 grams of dough for each cookie. Alternatively, use about 3 tablespoons of dough per piece.
Bake for 16–18 minutes, stopping when the edges appear set but the centers remain soft. Do not bake beyond this stage.
For a flatter finished shape, carefully drop or tap the hot baking sheet against a protected countertop once or twice immediately after removing it from the oven.
Leave the baked cookies on their baking sheets for several minutes so they can begin to set.
Transfer the cookies to a wire cooling rack.
Drizzle the reserved apple cider caramel glaze over each cookie. If desired, finish with a small amount of flaked sea salt.
Allow the caramel topping to set for several minutes before serving the cookies while they are still slightly warm.