Combine the broth, water, mirin, and the fish sauce in a large saucepan. Cut all but one scallion into 1-inch pieces and add to the broth with the mushroom stems, ginger, garlic and bunch basil. Bring to a simmer over medium-high heat and continue to simmer until reduced by half, about 45 minutes. Remove the solids from the broth using a strainer or slotted spoon and discard.
Add the chicken, sliced mushroom caps, and jalapeno to the broth; adjust heat to keep the broth at a gentle simmer and cook 12 to 14 minutes or until the chicken is just cooked through. Transfer the chicken to a cutting board and slice into 1-inch pieces.
Remove the broth from the heat; Stir in the edamame, jicama, and sliced chicken. Season to taste with salt. Scoop 1/2 cup of the cooked rice in each bowl and ladle some of the soup over it. Slice the reserved scallion on an angle and sprinkle over each bowl along with the basil leaves. Serve with lime wedges, if desired.
Jalapenos vary in heat, so adjust the amount of jalapeno you use to the spiciness of the pepper and your preferences.
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