Peel and dice the potatoes and carrots into approximately 1/2-inch (1.3 cm) pieces. Dice the onion and celery, mince the garlic, and cut the chicken into evenly sized 1/2-inch (1.3 cm) pieces. Keep the raw chicken separate from the vegetables and use clean equipment when switching between them.

Heat the olive oil in a 5- to 6-quart (4.7- to 5.7-L) Dutch oven over medium heat. Add the diced onion, carrots, and celery. Cook for 5 to 6 minutes, stirring occasionally, until the onion looks translucent and the vegetables begin to soften. The carrots should still feel slightly firm.

Stir in the minced garlic, dried thyme, and crushed rosemary. Cook over medium heat for 30 seconds, stirring constantly, until the garlic smells fragrant. Don't let the garlic turn dark brown, as it can develop a bitter flavor.

Sprinkle the flour over the softened vegetables: Stir continuously over medium heat for 1 minute until the flour coats the vegetables and no dry patches remain. The mixture should look slightly pasty, which is normal.

Slowly pour in the chicken broth while stirring to dissolve the flour and prevent lumps. Add the diced potatoes, salt, and black pepper. Increase the heat to medium-high until the mixture reaches a gentle boil, then reduce to medium-low. Cover partially and simmer for 8 to 10 minutes, stirring occasionally, until the potatoes begin to soften but still hold their shape.

Add the diced chicken to the simmering soup and stir to separate the pieces. Cover partially and continue cooking over medium-low heat for 8 to 10 minutes, stirring occasionally. The chicken should look opaque throughout and reach an internal temperature of 165°F (74°C) on an instant-read thermometer. The potatoes should also be fork-tender.

Reduce the heat to low and slowly stir in the cold milk. Add the frozen peas and cook gently for 3 to 4 minutes, stirring occasionally, until the peas are hot and bright green and the soup is steaming. Keep the soup below a vigorous boil to help the milk stay smooth.

Stir in the chopped parsley and continue heating over low heat for about 1 minute. The finished soup should be creamy enough to lightly coat a spoon, with tender vegetables and evenly distributed chicken. Ladle into six serving bowls and serve hot.