At the end of your time in Luncheon Kingdom, you will have to battle Cookatiel with Mario and Cappy. The battle with it will start with the bird launching food items at Mario (still as a captured Lava Bubble from earlier in the story). Avoid them to prevent damage, as well as receive more Coins and health that can be gathered as you do. Eventually, Cookatiel will shoot a wave of lava from its mouth. Use your Lava Bubble powers to surf up the lava and get close to Cookatiel. After you do, jump up and land of its face to hurt it.
As you drop back down into the pot of stew, Cookatiel will start launching more food in greater quantities at you. Once the bird starts shooting lava from its mouth, this time in the form of zigzags, swim up the path and land another blow to its face. The third and final phase of the battle will continue to see Cookatiel launch food at you before spitting more lava. The lava this time will be down in small burst, causing you to jump from one to the next as if they were platforms. Once you get close enough to the bird, land one more blow to defeat it and send it into the deep fryer.