Place the sugar and water for the caramel sauce into a medium heavy-bottomed saucepan. Gently swirl the pan so the sugar is completely moistened.

Set the pan over medium heat, swirling lightly every minute or so until the sugar dissolves. Once dissolved, bring it to a simmer, lower the heat, and cover with a lid. Let it cook until the syrup turns a pale amber shade, about 10 minutes, checking the color occasionally and swirling the pan as needed. Do not stir. If crystals appear on the sides, dissolve them with the steam or a damp pastry brush.

As soon as the caramel reaches a light amber color, remove the pan from the heat and stir in the softened butter. Slowly add the heavy cream while stirring continuously until the mixture is smooth. Transfer the caramel to a heatproof jar, then stir in the sea salt and vanilla extract.

Preheat the oven to 350°F for convection or 325°F for a conventional oven. Line a large baking sheet with parchment paper.

In a medium bowl, whisk together the flour, baking powder, cinnamon, and sea salt until evenly combined. Set this mixture aside.

Add the pecans and roughly one-third of the flour mixture to a food processor. Pulse until the nuts are finely ground.

In a large bowl, whisk the melted butter and brown sugar together until fully blended. Add the egg and vanilla, then continue whisking for 1 to 2 minutes, until the mixture looks smooth and slightly lighter in color.

Fold in the remaining dry ingredients, the ground pecan mixture, and the grated carrots. Mix until the dough is evenly combined.

Using a 2-ounce cookie scoop, portion 2.4 ounces of dough onto the prepared baking sheet. Bake for 8 minutes.

Remove the tray from the oven and firmly tap it on the counter to create ripples. Shape the cookies with a round biscuit cutter or a spatula if needed. Return them to the oven and continue baking for 6 to 8 minutes more, until the edges are lightly golden and the centers no longer appear wet.

Take the cookies out of the oven and once again tap and reshape them if desired. Let them rest on the baking sheet for 5 to 10 minutes, then transfer them to cool completely.

To prepare the frosting, beat the softened butter and cream cheese in a large bowl, or in the bowl of a stand mixer fitted with the whisk attachment, on medium-high speed for 1 to 2 minutes until smooth and well incorporated.

Add the powdered sugar and the pinch of sea salt. Begin mixing on low speed, then increase to medium-high and beat for another 1 to 2 minutes. Add the vanilla bean paste and enough heavy cream to reach the desired consistency, then beat for about 1 minute more.

Once the cookies are fully cooled, spread the cream cheese frosting over the tops. Finish each one with a drizzle of the salted caramel sauce.