Start by placing the whole fresh chicken, halved yellow onions, and garlic into a large stock pot or Dutch oven. Don't forget to add the bay leaves, jalapeno, kosher salt, and peppercorns. Pour in 12 cups of water until everything is submerged.
Bring the mixture to a simmer over medium heat. As it heats up, you'll notice some cloudy material rising to the top. Use a fine-mesh strainer to remove this scum, ensuring your broth remains clear and flavorful.
Once you've achieved a gentle simmer with small bubbles forming around the edges, lower the heat to medium-low and cover the pot with a lid. Allow the soup to simmer for about 1 hour and 30 minutes, or until the chicken becomes tender and starts to fall apart.
When the chicken is ready, carefully remove it and place it on a cutting board. Discard the onions, garlic, bay leaves, and peppercorns, making sure to retain the delicious broth.
For an optional step, refrigerate the broth for a few hours or overnight. This allows the fat to solidify on the surface, making it easy to scoop off the next day.
Return the broth to the pot and bring it back to a simmer over medium heat. Now it's time to add your Yukon Gold potatoes, sweet potatoes, and carrots. Cook them until tender, which should take about 20 to 30 minutes.
While the vegetables are cooking, shred or dice the chicken into bite-sized pieces, discarding the skin and bones. You can measure out 2 to 3 cups of the chicken to add back into the soup, saving any extra for another recipe.
Once the vegetables are soft, stir in the shredded chicken and let it simmer for an additional 2 to 5 minutes to warm through.
Finally, remove the soup from the heat and add the juice of the large lemon. Stir everything together and taste. Adjust the seasoning with salt and pepper as desired.