Walkthrough: The battle with Cookatiel will begin with the creature launching food items at Mario (still as a captured Lava Bubble). Avoid them not only to prevent damage, but also receive more Coins and health that can be gathered. Eventually, Cookatiel will spew a giant blast of lava from its mouth. Use your Lava Bubble powers to surf up the lava and get close to Cookatiel. As you do, jump up and land of its face to deal some massive damage to the bird.
As you drop down, Cookatiel will start launching more food in greater quantities at you. Once the bird starts spewing lava from its mouth, this time in the form of zigzags, swim up the path and land another blow to its face. The 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. You will earn a Multi Moon for winning.
This will end your time in Luncheon Kingdom. If you still are needing a few more Power Moons before you can leave, start exploring the kingdom in order to find them. After you have enough Power Moons in order to leave, hop in your Odyssey Airship. The plan will be to now move forward to Bowser Kingdom; that is, until Bowser shows up with a new winged friend and sends you hurling into Ruined Kingdom.