Brothy Chicken and Rice with Ginger and Scallions

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6 cups reduced-sodium chicken broth 

2 cups water 

2 tablespoons mirin 

1 tablespoon fish sauce 

1 bunch scallions (about 6) 

4 ounces shiitake mushrooms, caps thinly sliced, stems reserved 

1 (2-inch) piece fresh ginger, peeled and sliced 

6 garlic cloves, sliced 

1 small bunch basil, plus basil leaves, for serving 

1 large chicken breast (about 12 ounces) 

1/2 to 1 red jalapeno, thinly sliced (seeds and ribs removed for less heat, if desired) 

1 cup frozen shelled edamame, thawed 

3 ounces (1 cup) julienned jicama 

Kosher salt 

2 cups hot cooked brown rice 

Lime wedges, serving suggestion