Cover the chicken breast with plastic wrap and gently pound it to an even thickness. Season all over with salt and pepper. Chop the bacon into small pieces. Peel the potatoes and cut them into small cubes about 1 cm in size.
Heat a large, wide pot or Dutch oven over medium to medium-low heat. Cook the bacon, stirring occasionally, until it reaches your desired level of doneness. Remove to a paper towel lined plate and leave the drippings in the pan.
Increase the heat to medium-high, ensure you have enough bacon grease to full coat the bottom of the pot. If not, supplement with a little bit of olive oil. Once hot, add the chicken breast and sear on both sides until golden, about 2-3 minutes per side. Remove the chicken to a plate, but do not wipe the pan clean. (The chicken does not need to be cooked through)
Evaluate how much bacon grease remains in the pot. If it doesn't fully coat the bottom of the pot, supplement with a little bit of olive oil. Reduce the heat to medium and cook the onions in the bacon drippings until they soften, about 5 minutes, stirring occasionally.
Add potatoes, garlic, gochujang paste, salt, and pepper to the pot. Cook for about 5 minutes, stirring frequently.
Dust the flour over the mixture in the pot and stir until the ingredients are well-mixed and starting to take on more color, ~2-3 minutes.
Add the half-and-half and water and stir until well-mixed. Bring to a boil over high heat, and then reduce heat to medium-low and simmer. Add the chicken breast back into the soup once simmering to finish cooking.
The outsides of the potatoes will dissolve into the liquid, creating the rich creamy texture with small chunks of potato remaining. The longer you simmer, the more the potatoes will dissolve into the soup. I simmer for at least 30 minutes but sometimes longer. Stir occasionally to prevent potatoes from sticking to the pan's bottom.
Let the chicken breast simmer in the soup until it's completely cooked through, about 15 minutes. Then remove it to a cutting board and shred (I do this using two forks). Tent with foil to keep warm.If you have a meat thermometer, you can check to make sure the internal temp of the chicken has hit 165 °F
Before serving, add more salt and pepper to taste. Potatoes absorb a ton of salt, so it may feel like you are over-salting, but you probably aren’t. Add the shredded chicken back into the soup, mix, then remove from heat. Ladle into bowls and serve topped with shredded cheddar, green onion, and bacon bits.
Notes
Can be prepared without the bacon, just cook the onions in ~2 tablespoon veggie oil.