Fill a medium stock pot with water and bring your chicken breast to a boil. Reduce the heat to medium high and cook for about 20- 30 minutes. Remove the chicken and shred it to bite size pieces.
In a medium stock pot, melt 1 1/2 tablespoons butter over medium-high heat. Add in onion, carrot and celery and sauté 3 minutes.
Add chicken broth, potatoes, parsley, thyme, rosemary, bay leaf and season with salt and pepper to the pot. Bring soup to a boil, then reduce heat to medium, cover with lid, and allow to cook. Making sure you stirring occasionally.
Once the potatoes are tender, about 15 - 25 minutes. Reduce heat to low and stir in chicken and peas. Remove the thyme twigs and bay leaf.
In a medium saucepan set over medium heat, melt remaining 4 tablespoons butter. Stir in flour and cook mixture, stirring constantly for 1 1/2 minutes.
While whisking very slowly pour in milk and whisk vigorously to smooth lumps. Season with salt and pepper to taste.
Bring mixture to a boil over medium high heat while stirring constantly, then remove from heat and stir in cream.
Stir the cream mixture into the soup.