Cream the butter and sugars. In a large bowl, beat the room temperature butter, granulated sugar, and brown sugar with a hand mixer or stand mixer for 2 to 3 minutes, until smooth, pale, and fluffy.
Add the eggs and vanilla. Add the eggs and vanilla extract and beat until everything is creamy and fully combined.
Whisk the dry ingredients. In a separate bowl, whisk together the flour, baking powder, cornstarch, and salt. Whisking first spreads the baking powder and salt evenly through the dough.
Combine. Add the dry ingredients to the wet ingredients and mix on low or fold with a spatula just until no dry flour remains. Don't overmix, or the cookies will be tough.
Chill the dough. Cover the bowl with plastic wrap and refrigerate for at least 30 minutes. This keeps the cookies from spreading too much and gives them that thick, bakery-style texture.
Make the caramel while the dough chills. Place a small heavy-bottomed saucepan over medium heat and add the sugar. Stir constantly with a heatproof spatula. It will clump up and look like it's going all wrong, but keep stirring! In about 4 to 5 minutes, it will melt into a smooth, golden amber liquid. Watch it closely and lower the heat if it's browning too quickly.
Add the butter. Take the pan off the heat and stir in the cubed butter. It will bubble up a LOT, which is totally normal. Keep stirring until the butter is completely melted and smooth.
Add the cream and salt. Slowly pour in the heavy cream while stirring. It will bubble again, so go slow. Stir until creamy and smooth, then stir in the fine sea salt. Pour the caramel into a heatproof jar and let it cool while you bake. It thickens as it cools into the perfect drizzling consistency.
Preheat the oven. Preheat your oven to 375°F (190°C) and line your baking sheets with parchment paper.
Scoop and bake. Scoop the chilled dough into balls with a medium cookie scoop and place them about 2 inches apart. Bake for 10 to 12 minutes, until the edges are lightly golden but the centers still look soft. Pull them out a little early for the softest cookies!
Cool completely. Let the cookies cool on the baking sheet for 5 minutes, then move them to a wire rack to cool completely, about 20 minutes. They need to be fully cool before frosting, or the frosting will melt.
Whip the frosting. In a large bowl, beat the cream cheese and butter for 2 to 3 minutes, until smooth and fluffy. Add the vanilla, then add the powdered sugar ½ cup at a time, starting on low speed so it doesn't fly everywhere. Beat until thick, smooth, and creamy. Too thick? Add a splash of milk. Too thin? Add a little more powdered sugar.
Frost the cookies. Spread a thick layer of frosting on each cooled cookie with an offset spatula, or pipe it on for a prettier look.
Drizzle and sprinkle. Stir the caramel. If it's too thick to drizzle, microwave it for 10 to 15 seconds. Drizzle it over the frosted cookies with a spoon or piping bag, then sprinkle with a little flaky sea salt while the caramel is still warm. Let it set for a few minutes, then ENJOY, besties!