Grate apples on the large holes of a box grater or in a food processor with the grater attachment. Transfer grated apples to a clean kitchen towel and squeeze over a medium bowl to extract all the juice possible. Measure out ¼ cup apple juice and set aside (save any leftover apple juice for another use). Set grated apples aside (you should have about 2½ cups).
Bring sugar, cream of tartar, and reserved ¼ cup apple juice to a rapid boil in a large saucepan over medium heat, stirring just to dissolve sugar. Cook, without stirring but swirling pan often, until bubbles slow and caramel turns a deep amber color, 5–7 minutes. Remove caramel from heat and stir in butter, salt, cinnamon, cardamom, and nutmeg.
Set pan over medium-low heat and add reserved grated apples, stirring quickly to combine (don’t worry if the caramel mixture looks like it’s seizing at first). Cook apple caramel, stirring constantly, until it’s noticeably stickier, forms a ball as you stir, and pulls away from the sides of the pan in long strands, 10–12 minutes. Scrape apple caramel onto a parchment-lined baking sheet and spread out in an even layer. Let cool, at least 30 minutes. Set ¼ cup apple caramel aside for topping cookies.
Place racks in upper and lower thirds of the oven and preheat to 375°. Whisk flour, salt, baking soda, and cream of tartar in a medium bowl to combine.
Beat butter, granulated sugar, and brown sugar in a stand mixer fitted with the paddle attachment, starting on low speed, then increasing speed to medium, until combined, about 2 minutes total. Add eggs one at a time, incorporating thoroughly before adding the next, then add vanilla. Reduce speed to low; add dry ingredients and mix just until combined. Add apple caramel and pulse mixer two or three times to swirl caramel throughout dough or fold in caramel using a sturdy rubber spatula (take care not to overmix or else you’ll lose the swirled caramel effect as the cookies bake).
Using a #20 ice cream scoop (about 3 Tbsp.), portion out 16 balls and divide between 2 parchment-lined baking sheets, spacing at least 2" apart (you will have dough left over). Top each cookie with a few small pieces of reserved apple caramel and sprinkle with sea salt.
Bake cookies, rotating baking sheets top to bottom and front to back halfway through, until golden brown at the edges, 13–16 minutes. Let cookies cool for 5 minutes on baking sheets then transfer to a wire rack and let cool completely. Let 1 baking sheet cool, then repeat the process with remaining dough to make 8 more cookies (they may bake slightly faster). Do ahead: Cookies can be made 5 days ahead. Store airtight at room temperature.