Place apple butter into a fine mesh strainer set over a bowl. Allow apple butter to drain while preparing dough. The apple butter should reduce to about 1/2 cup, with about 1/4 cup juice strained out. Reserve juice to use for icing.

Place unsalted butter into a light-colored skillet over medium-low heat. Cook, stirring frequently, until little browned bits of butter begin to form and stick to the bottom of the pan, 5 to 10 minutes. Immediately pour browned butter into a heat-safe bowl and set aside to cool for 15 minutes.

Meanwhile, place oats into the bowl of a food processor. Pulse until oats are slightly finer in texture, but not powdered, 12 to 14 times. Set aside.

Peel and grate Granny Smith apple to yield about 1 cup loosely packed grated apple. Place the grated apples onto some paper towels and squeeze gently to remove some of the excess moisture, until you have 1/2 cup firmly packed grated apple. Set aside.

To cooled browned butter, add brown sugar, white sugar, cinnamon, salt, nutmeg, and allspice and whisk until thoroughly combined. Add in strained apple butter, egg yolks, 2 teaspoons vanilla, and grated apple. Whisk until completely incorporated. Add in oats, flour, cornstarch, and baking soda, and mix until combined. Cover bowl, and place into the refrigerator to chill for 30 minutes.

Preheat the oven to 350 degrees F (180 degrees C). Line baking sheets with parchment paper.

Remove dough from refrigerator. Scoop out 1 1/2 tablespoon-sized balls of dough and drop them 2 inches apart on the prepared baking sheet. Flatten cookies to about 1/2-inch thickness using your fingers (if dough sticks, slightly dampen your hand with a little water before pressing the cookies down).

Bake cookies in the preheated oven until they appear just set and turn slightly darker in color, 10 to 12 minutes. Cool cookies on baking sheets for 5 minutes, then carefully remove to a wire rack to cool completely.

For icing, whisk together reserved 1/4 cup juice from apple butter, confectioner’s sugar, salt, cinnamon, vanilla, and heavy cream until smooth and combined, thin but not watery, about as thick as honey. Add heavy cream, 1 teaspoon at a time, to thin as needed. If too thin, add confectioner’s sugar, 1 tablespoon at a time, to thicken icing.

Hold cookies by their bottom edges, and dip tops of cookies lightly into icing--just barely touching icing. Repeat dipping until all cookies are iced. Allow icing to set before storing.