Lightly spray 9-inch round, 2-inch deep cake pan with nonstick cooking spray; set aside.
Combine pineapple, brown sugar, and ginger in 10-inch skillet; cook over medium heat, stirring occasionally during first 5 minutes, until pineapple is translucent and has light brown hue, 15 to 18 minutes. Empty fruit and juices into mesh strainer or colander set over medium bowl. Return juices to skillet, leaving pineapple in strainer (you should have about 2 cups cooked fruit). Simmer juices over medium heat until thickened, beginning to darken, and mixture forms large bubbles, 6 to 8 minutes, adding any more juices released by fruit to skillet after about 4 minutes. Off heat, whisk in butter and vanilla; pour caramel mixture into prepared cake pan. Set aside while preparing cake. (Pineapple will continue to release liquid as it sits; do not add this liquid to already-reduced juice mixture.).
Adjust oven rack to lower-middle position and heat oven to 350 degrees. Whisk flour, baking powder, and salt in medium bowl; set aside.
In bowl of standing mixer fitted with flat beater, cream butter, sugar and coconut at medium-high speed until light and fluffy, 3 to 4 minutes. Reduce speed to medium, add vanilla, and beat to combine; one at a time, add whole eggs then egg white, beating well and scraping down bowl after each addition. Reduce speed to low; add about one-third of flour mixture and beat until incorporated. Add half of coconut milk and beat until incorporated; repeat, adding half of remaining flour mixture and remaining coconut milk, and finish with remaining flour. Give final stir with rubber spatula, scraping bottom and sides of bowl to ensure that batter is combined. Batter will be thick.
Working quickly, distribute cooked pineapple in cake pan in even layer, gently pressing fruit into caramel. Using rubber spatula, drop mounds of batter over fruit, then spread batter over fruit and to sides of pan. Tap pan lightly against work surface to release any air bubbles. Bake until cake is golden brown and toothpick inserted into center of cake comes out clean, 45 to 50 minutes. Cool 10 minutes on wire rack, then place inverted serving platter over cake pan. Invert cake pan and platter together; lift off cake pan. Cool to room temperature, about 2 hours; then cut into pieces and serve.