It is first seen when Rusty announces he's leaving home to become a trainer, sitting on the table of Rusty's house. Rusty's mom asks him to bring it to Professor Tree. Instead of delivering it to the Professor, he took it along with his journey. It was unknown if he actually ate it.
In Bidocalypse, it was most likely that the Chicken Panini was never delivered to Prof. Tree once again by Future Rusty.