Put the chicken in a medium Dutch oven or heavy pot and add the bell pepper and chicken broth. Bring to a boil. Cover and reduce the heat to medium low; gently simmer until the chicken is just cooked through, about 12 minutes. Transfer the chicken and bell pepper to a plate using a slotted spoon; let cool. Shred the chicken.
Add the beans to the pot with the broth. Increase the heat to high, cover and cook until crisp-tender, 7 minutes. Transfer the beans to a colander using a slotted spoon, reserving the broth in the pot; rinse under cold water and let drain. Transfer to a medium bowl and toss with the lemon juice, 1 tablespoon olive oil and 1/4 teaspoon salt.
Meanwhile, lightly toast the almonds in a dry skillet over medium heat, 1 to 2 minutes; remove to a plate and set aside. Add the remaining 1 tablespoon olive oil and the garlic to the skillet. Cook until softened, about 1 minute. Add the bread. Cook, stirring often, until golden, about 5 minutes. Transfer the garlic and bread to a blender or food processor along with the bell pepper, 1/2 cup of the warm broth, 1 tablespoon each capers and parsley, and 1/4 teaspoon salt. Puree until smooth; transfer to a large bowl.
Add the chicken to the red pepper sauce; season with salt and toss. Top each serving with the remaining capers and parsley; serve the green beans on the side, sprinkled with the almonds.