Place a rack in middle of oven; preheat to 325°. Lightly coat sides of a 13x9" metal baking pan with nonstick vegetable oil spray. Line bottom of pan with parchment paper. Toast 1½ cups (120 g) pecans on a small rimmed baking sheet until fragrant and deep brown, 12–18 minutes. Let cool; coarsely chop. Increase oven temperature to 375°.

Vigorously whisk 2 cups (250 g) bleached all-purpose flour and 2 tsp. baking powder in a large bowl to combine.

Blend 3 small carrots (10.5 oz.), scrubbed, trimmed, cut into 1" pieces, 3 large eggs, room temperature, 2 cups (packed; 400 g) light brown sugar, 1 cup vegetable oil, 1 Tbsp. ground cinnamon, 2 tsp. Diamond Crystal or 1¼ tsp. Morton kosher salt, and 2 tsp. vanilla bean paste or vanilla extract in a blender until very smooth, about 2 minutes. Add carrot mixture to bowl with dry ingredients and whisk until just combined (some streaks of dry flour should still be visible). Using a rubber spatula, fold in pecans, 1 cup (140 g) finely chopped dried pineapple rings and ¼ cup (45 g) finely chopped crystallized ginger, scraping down sides and bottom of bowl as needed, until well combined. Scrape batter into prepared pan and jiggle to level surface.

Bake cake until golden brown and top springs back when lightly pressed (an instant-read thermometer inserted into the center should register 200°), 25–30 minutes. Transfer pan to a wire rack and let cake cool, about 1 hour. If you prefer to serve the cake unmolded, invert cake onto rack and peel away parchment, then invert cake, right side up, onto a platter.

Using an electric mixer on medium speed, beat 8 oz. cream cheese, room temperature, 2 tsp. vanilla bean paste or vanilla extract, and 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt in a medium bowl until smooth, about 3 minutes. Add 2 cups chilled heavy cream and ⅓ cup (67 g) granulated sugar; beat until frosting holds firm peaks, about 5 minutes. Scrape down sides and bottom of bowl; fold gently to ensure mixture is well combined.

Spread frosting over top of cake all the way to the edges with an offset spatula. Using a spoon or offset spatula, create decorative swooshes and swirls in frosting if desired. Dust lightly with ground cinnamon. Do ahead: Cake can be assembled 1 day ahead. Cover and chill. Bring to room temperature before serving.