Preheat the oven to 350. Add cold, cubed butter to a large bowl or stand mixer. Add white and brown sugar and mix for 2-3 minutes, until butter is fully incorporated.
Add eggs, greek yogurt, and vanilla and mix until just combined.
Add flour, cornstarch, baking soda, and salt. Mix until just combined.
Using a large cookie scoop (about 1/4 cup), scoop out the dough onto a lined or greased cookie sheet. Using the back of the cookie scoop or a spoon, make a small crater on the top of each cookie -- this is where the apples will go.
Peel the apples and chop them into small pieces. Toss the apples with the cinnamon until they're well coated.
Use a spoon to scoop about 1 tbsp of the cinnamon apples into the crater on each cookie. The apples shouldn't overflow off the cookie, but should be as full as you can get them.
Bake the cookies for 10-12 minutes, until they're just set. They may look slightly undercooked in the center underneath the apples, but this is because of the apple juices. If the edges look set, they're done. Let them cool on the pan for at least 5 minutes then transfer them to a cooling rack.
While the cookies cool, make the frosting. To brown the butter add it to a small pot over medium heat. Stir the butter continuously as it melts. It will begin to bubble, then after a couple minutes it will start to foam up. When the foam goes down, you should be left with brown, nutty smelling butter. This should take 4-7 minutes.
Once the butter is browned, take it off the heat and pour it into a bowl. Transfer the bowl to the freezer for 10 minutes so the butter isn't so hot.
Add the cooled brown butter to a large bowl with the cream cheese. Mix for 2 minutes. Add the powdered sugar, maple syrup, and vanilla. Mix another 2 minutes. The frosting will smell incredible and taste even better!
Transfer the frosting to a piping bag with a flat tip. Pipe the frosting onto each cookie, three horizontal stripes and three vertical stripes. Enjoy!
Since these cookies have fresh apples on them, they are best enjoyed within 1-2 days of baking them. After that the juice from the apples starts to make the cookies soggy.