Bring 1 1/2 cups water to a simmer in a medium skillet. Whisk in the peanut butter, then add the chicken. Cook over medium-low heat at a bare simmer until just cooked through, 3 to 5 minutes. Use a slotted spoon to transfer the chicken to a plate. Continue to simmer the sauce until thickened and reduced by about half, about 5 minutes. Take off the heat and whisk in the vinegar, soy sauce, Sriracha, and scallions. Cool slightly.
Add the rice noodles to the boiling water and cook until almost tender, about 3 minutes. Drain and rinse under cold water to cool. Drain the noodles very well, tossing with tongs to release as much water as possible. Snip the noodles with scissors a few times to cut the long strands.
Toss the noodles with the romaine, cucumbers, bell pepper, half of the cilantro, and just enough dressing to coat. Season with salt. Divide among 4 bowls and top with the chicken. Drizzle the remaining dressing over the top and garnish with the remaining cilantro. Serve with lime wedges.
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