Set the oven to 350°F (175°C). Lightly grease a 9 × 13-inch baking pan and line the bottom with parchment paper if desired. Allow the oven to finish heating before the batter goes in so the cake begins baking at the intended temperature.

Place the eggs and sugar in a large bowl. Whisk until the mixture looks smooth and slightly lighter. Pour in the vegetable oil and vanilla extract, then mix until evenly blended. Add the drained pineapple and prepared carrots, stirring gently so both are distributed through the liquid mixture.

In another bowl, combine the flour, baking powder, baking soda, and cinnamon. Stir with a whisk to break up any pockets of leavening and spread the cinnamon evenly through the flour.

Gradually add the dry mixture to the bowl containing the pineapple and carrot mixture. Fold with a rubber spatula just until no obvious streaks of flour remain. Avoid prolonged mixing, which can make the cake less tender. Add the chopped pecans and fold them through at the end.

Scrape the batter into the prepared pan. Use the spatula to spread it toward the corners and level the surface. Give the pan a gentle tap against the counter to release any large trapped air pockets.

Place the pan on the center oven rack and bake for approximately 40 to 45 minutes. Begin checking around the 40-minute mark. The center should look set, and a toothpick inserted into the middle should come out with a few moist crumbs rather than wet batter.

Set the pan on a wire rack and let the cake cool completely. Do not spread cream cheese frosting over a warm cake because the heat can soften the frosting and make it slide. Once cooled, gently loosen the edges if needed before frosting.

Beat the softened cream cheese and butter until creamy. Add the powdered sugar gradually, mixing after each addition. Blend in the vanilla extract until the frosting is smooth. Spread it evenly over the cooled cake with an offset spatula. Slice and serve once the frosting has settled.