OKAY so I think I've narrowed down all the Zukaang episodes based on the parallels, meaningful interactions, and near misses, highlighting their connection and role as foils and soulmates. Book 1 obviously has a lot of Zuko trying to chase and capture Aang, but for the purposes of this post I am not including those.
Episode 12, "The Storm" - Direct parallels between Zuko and Aang, delving into their separate backstories and recollecting the moments that changed their lives forever.
Episode 13, "The Blue Spirit" - Aang is captured by Zhao and Zuko, disguised as the Blue Spirit, saves him. They work together to fight their way out of Zhao's prison. Aang asks Zuko if they could ever be friends.
Episode 19 & 20, "The Siege of the North" - Zuko finally manages to capture Aang. Zuko talks to Aang while Aang is unconscious. Zuko carried an unconscious Aang on his back and later Aang carried an unconscious Zuko on his back.
Episode 9, "Bitter Work" - Both Aang and Zuko continue their training and both struggle as Aang tries to learn earthbending and Zuko tries to learn lightning and lightning redirection. While Aang ends up successful in his attempts, Zuko is not.
Episode 12, "The Serpent's Pass" - Both Aang and Zuko travel to Ba Sing Se. While Aang and the Gaang go through the treacherous, titular Serpent's Pass to accompany some refugees who lost their tickets, Zuko and Iroh get to go on the ferry that the Gaang would've been on. Both Zuko and Aang are feeling terrible during their travels; Aang is mourning Appa and Zuko hates having to live as a refugee. It ends with Zuko teaming up with Jet, and Aang having hope again.
Episode 17, "Lake Laogai" - Both Aang and Zuko search for Appa and get to Lake Laogai, but very narrowly miss each other. Zuko gets there first, and frees Appa. I could also argue that Jet connects them both as well, considering Zuko is the reason why Jet was captured and brainwashed by the Dai Li in the first place.
Episode 1, "The Awakening" - Aang and Zuko both journey on separate ships headed for the Fire Nation. While Aang believes he has lost his honour, Zuko believes he has regained it.
Episode 6, "The Avatar and the Fire Lord" - Aang learns the story of his past life, Roku, while Zuko learns the story of his great-grandfather, Sozin. Roku and Sozin were best friends whose lives and destinies were so entwined that they shared a birthday. Their bond prevented Roku from killing Sozin, leading to the war that only Aang and Zuko can resolve. It is also revealed that Roku is also Zuko's great-grandfather on his mother's side, further emphasising how Zuko and Aang's lives and destinies are intertwined.
Episode 10 & 11, "The Day of Black Sun" - Aang and Zuko both ready themselves to face Fire Lord Ozai. Zuko gets there first while Aang is fighting Azula. This also parallels the confrontations that will occur in the finale.
Episode 12, "The Western Air Temple" - Aang and Zuko both arrive at the Western Air Temple. While Aang has fond memories of this place, Zuko has bad ones. Aang needs a firebending teacher and Zuko has come to offer himself as one.
Episode 13, "The Firebending Masters" - Self-explanatory. Aang and Zuko go on their field trip to the ancient Sun Warrior city. They perform the Dancing Dragon in front of Ran and Shaw, both relearning firebending together. Ran and Shaw circle and dance around each other, representing taegeuk, much like how Tui and La's eternal dance represents yin and yang.
Episode 18-21, "Sozin's Comet" - Both Aang and Zuko face off against their most dangerous opponents yet in separate 1v1 epic battles to determine the fate of the world. Aang vs Ozai, Zuko vs Azula. This parallels the confrontations during The Day of Black Sun where Aang faced Azula and Zuko faced Ozai. Both their victories end the war, ushering a new era of peace, and cementing their roles as the Avatar and the Fire Lord, a direct parallel to Roku and Sozin who had started this war.
In conclusion: Whatever souls are made of, Aang's and Zuko's are the same ¯\_(ツ)_/¯
