One could start by looking on popular fanfiction platforms. Some might be about Aang's journey after being separated from his friends for a long time. Maybe there are stories where he has to face new challenges alone in a strange land.
Well, often in aang abandoned fanfiction, the plot could revolve around how Aang copes with the feeling of abandonment. He might have to rediscover his powers or find new allies. There could be a story where he stumbles upon a hidden civilization that helps him regain his confidence.
One possible fanfiction could be where Aang sacrifices himself to save the world from a new, powerful dark spirit. In this story, his friends are left to carry on his legacy of peace and balance.
One great female Aang fanfiction could be one where she is a powerful waterbender from the Southern Water Tribe. She meets Aang during his journey and they form a strong bond as they fight against the Fire Nation together. It could explore their friendship and how she helps him master waterbending techniques.
Some fanfictions focus on the emotional side of their relationship. For example, Aang helps Toph deal with her family issues while Toph teaches Aang how to be more assertive. It's a really interesting exploration of their characters in a non - combat - related scenario.
One good 'Aang broken' fanfiction could be one where Aang struggles with the aftermath of a great battle. Maybe he has lost some of his friends and is dealing with survivor's guilt. This could lead to him being 'broken' in the sense of his spirit being damaged.
One popular 'Aang and Katara romance fanfiction' is 'The Journey of Their Hearts'. It beautifully depicts their relationship development after the end of the series. Another great one could be 'Aang and Katara: Love Beyond the Elements' which focuses on how they overcome cultural differences in their love. There are also many on fanfiction.net, you just need to search for 'Aang and Katara romance' and explore the ones with high ratings.
One popular one might be 'Aang and Toph's First Kiss' which beautifully captures the moment with great description of their emotions. It shows how their relationship has developed to that point.
Some fanfictions in this category could focus on Aang's relationships. For example, a story about how Aang helps Toph deal with a family issue that she's been hiding. This would show a different side of Aang, not just as the Avatar but as a true friend. It could also explore how their friendship deepens through this experience.
You can try Archive of Our Own (AO3). It has a vast collection of fanfiction from various fandoms, and there's a good chance you'll find aang abandoned fanfiction there.
One great example could be a fanfiction where Aang is meditating to connect with his past lives in more detail. He might be seeking wisdom from Avatar Roku during a time of great crisis in the story. Another example could be a story where Aang meditates to find inner peace after a difficult battle. This helps him regain his strength and focus on the next steps for his journey.
In some fanfictions, Aang's energybending could be explored in a more character - development - centered way. For instance, after a difficult battle, Aang realizes that his energybending has a connection to his inner peace. He then goes on a journey to better understand this connection and in the process, becomes an even more powerful and wise Avatar. He might learn that by maintaining a perfect balance within himself, his energybending can reach new heights and he can use it to heal not only physical wounds but also emotional and spiritual ones of those around him.