Make the cake: Heat oven to 350°. Line the bottoms of three 9" round cake pans with parchment paper. Grease each pan with 1 tbsp. butter and dust with a little flour; shake pans so that flour evenly coats them. Invert pans and tap out excess flour; set pans aside.
In a large bowl, sift together the flour, cinnamon, baking soda, and salt; stir in the pecans and set aside. Using a handheld mixer set on high speed, beat the sugar, brown sugar, and eggs in a large mixing bowl until smooth, 1 minute. Add the crushed pineapple and juice, bananas, oil, and vanilla and mix until combined. Add the pecan mixture and mix on low speed until evenly combined.
Divide the batter evenly between the 3 prepared cake pans, smoothing the tops with a rubber spatula. Bake cakes, rotating occasionally, until a toothpick inserted into the center of each cake comes out clean, about 50 minutes. Transfer cakes to racks and let sit for 15 minutes; invert pans and unmold each cake; discard parchment paper. Let cakes cool.
Make the frosting: In the bowl of a standing mixer fitted with a paddle, beat the butter on medium speed until smooth, 2 minutes. Add the cream cheese and vanilla and beat until creamy, 1–2 minutes more. Add the confectioners' sugar and mix on low speed until incorporated. Increase speed to medium-high and continue mixing until the frosting is light and fluffy, 2–3 minutes. Remove any dust from the pecans by shaking them in a wide-mesh sieve over the sink; using a rubber spatula, fold the pecans into the frosting.
Assemble the cake: Place 1 cake layer on a cake stand. Using a butter knife, spread 2⁄3 cup frosting evenly over the top. Place second cake layer atop first one and frost top of it with another 2⁄3 cup frosting. Repeat with remaining cake layer. Spread remaining frosting over the sides of the assembled cake. Serve immediately or wrap loosely with plastic wrap and refrigerate.