Heat 2 teaspoons oil in a large saucepan or Dutch oven over medium-high heat. Add carrot (or bell pepper) and chicken; cook, turning the chicken and stirring frequently, until the chicken begins to brown, 3 to 4 minutes. Add garlic and cook, stirring, for 1 minute more. Stir in broth and marjoram; bring to a boil over high heat. Reduce the heat and simmer, stirring occasionally, until the chicken is cooked through, about 5 minutes.
With a slotted spoon, transfer the chicken pieces to a clean cutting board to cool. Add spinach and beans to the pot and bring to a gentle boil. Cook for 5 minutes to blend the flavors.
Combine the remaining 1 tablespoon oil, Parmesan and basil in a food processor (a mini processor works well). Process until a coarse paste forms, adding a little water and scraping down the sides as necessary.
Cut the chicken into bite-size pieces. Stir the chicken and pesto into the pot. Season with pepper. Heat until hot. Garnish with croutons, if desired.