Set the oven to 350°F. Fit a 13x18-inch half-sheet pan with parchment paper, creasing the folds at the corners so it lies flat.

In a large mixing bowl, whisk the sugar, oil, eggs, baking powder, baking soda, salt, cinnamon, ginger, nutmeg, and cloves until smooth and well combined.

Gently stir in the flour just until blended, then fold in the grated carrots, toasted pecans, crushed pineapple, and shredded coconut. Transfer the batter to the prepared pan and spread it evenly.

Bake for 28 to 33 minutes, until a skewer inserted into the center comes out clean or with just a few moist crumbs. Leave the cake in the pan and let it cool completely.

Lower the oven to 300°F and line a rimmed baking sheet with parchment paper or a silicone baking mat.

In a medium bowl, beat the egg white until it is loose and foamy. Add the pecan pieces and toss to coat them evenly. Sprinkle in the granulated sugar, brown sugar, cinnamon, and salt, and stir until every piece is well covered.

Spread the pecans in a single layer on the prepared baking sheet. Bake for 40 minutes, stirring halfway through, until glossy, browned, and fragrant. They will become crisp as they cool, so let them cool completely before using.

In a small saucepan, whisk together the egg yolk, heavy cream, and brown sugar. If you have corn syrup, add it now to help discourage crystallization; it can be omitted. Cook over medium-low heat, stirring constantly, until the mixture reaches a boil. Let it boil for 2 to 3 minutes, until it thickens and bubbles. The constant stirring keeps the egg yolk from scrambling, but a little crystallization later is normal.

Take the pan off the heat and stir in the butter, vanilla, and a pinch of salt. Set it aside and let the sauce cool completely before layering the cake.

In a stand mixer fitted with the paddle attachment or in a large bowl with a hand mixer, beat the cream cheese and butter just until smooth, being careful not to overwhip. Gradually add the powdered sugar, about 1/2 cup at a time, beating only until incorporated. Add the salt and vanilla, then beat on medium-high for 1 to 2 minutes until pale and fluffy. For a firmer, sweeter frosting, use up to 4 cups of powdered sugar; for a softer spreadable consistency, beat in a tablespoon or two of milk.

Slice the cooled cake in half lengthwise, then in half crosswise, to create four equal rectangular pieces.

Place one rectangle on a serving plate and spread with a layer of cream cheese frosting. Scatter about a quarter of the candied pecans over the frosting, then drizzle generously with the salted caramel. Repeat with the remaining cake rectangles, stacking them with frosting, pecans, and caramel. Fill any gaps along the edges with extra pecans. Leave the sides bare for a rustic look or frost them if you prefer. Drizzle each slice with any leftover caramel and sprinkle with extra pecans before serving.