I notice the dish from their menu on the cafe's official website.
It seems that they serve tea-based pot soup, which sounds pretty unique. It's not traditionally Japanese, but sort of like a modern version of cha-zuke, and probably far from Korean dishes.
From their photo, we can guess that the tea is sen-cha, so you'd want to use sen-cha 煎茶 tea leaves to make the tea to pour in your soup.
Perhaps the rice is not cooked from scratch. Plain boiled rice is probably thrown into the pot along with the other ingredients and then simmered. So it's not stewed but simmered, and it's not just hot tea poured over cold ingredients as you would with a cha-zuke.
You apart from the steamed chicken, you at least need a plump version of ume-boshi 梅干し (dried plum), and raw myouga みょうが (Japanese ginger) which really does the trick. From the photo, it looks like there are shiitake mushrooms in there too. Hoshi-shiitake 干ししいたけ (dried shiitake) would bring great taste to the soup than raw ones. You need to soften hoshi-shiitake by keeping it in some water for half a day. I wonder if it's tororo-konbu とろろ昆布 (shredded kelp) on top, because that brings great taste too. With ume-boshi and tororo-konbu, you hardly need salt, but a little bit soy sauce might add to the taste.
So ume-boshi and tororo-konbu can be topped as-is. Myouga is sold as a whole herb on shelves you find vegetables at, so you need to slice them. Hoshi-shiitake should be softened and then cut into pieces. Here is a photo of myouga. You can also grow it in the tiniest of gardens.https://www.google.com/search?q=%E3%81%BF%E3%82%87%E3%81%86%E3%81%8C&s...
Again, there is no traditional recipe, so I'm just guessing, but the ingredients I guessed typically do go good together, and I hope you enjoy your original version!