To a large pot or Dutch oven, heat the olive oil over medium heat. Add the onion, sprinkle with ¼ teaspoon of the salt, and cook, stirring occasionally, until it begins to soften, about 3 minutes.
Add the carrot and celery, stirring to combine. Sprinkle with ¼ teaspoon of salt, then cook 3-5 minutes until slightly softened, stirring occasionally.
Add the tomato paste and garlic, and cook until fragrant, about 1 minute, stirring constantly.
Add the potatoes, diced tomatoes, bay leaf, stock, chicken, and 1 teaspoon of salt, stirring to combine.
Cover, bring to a boil, then reduce to a simmer. Cook, covered, for 20-25 minutes, or until the potatoes are tender and the chicken is cooked through.
Uncover and remove the chicken to a cutting board.
Add the green beans, corn, and peas to the pot, stirring to combine. Simmer on low for 5 minutes to warm the vegetable through and for the flavors to absorb.
While the veggies simmer, shred the chicken with two forks, or chop into bite-sized pieces.
Turn off the heat, then add back the chicken, stirring to combine. Taste and add additional salt if desired.