Preheat the Oven (Optional):
If preparing the toasted oyster crackers, preheat your oven to 375°F. Line a large rimmed baking sheet with foil for easy cleanup.
Sauté the Vegetables:
In a large Dutch oven, melt the butter over medium heat. Add the onion, carrots, celery, garlic, thyme, and salt. Cook, stirring frequently, until the vegetables become soft and fragrant, about 10–12 minutes.
Thicken the Base:
Sprinkle the flour over the vegetables and stir constantly for approximately one minute, ensuring the flour coats the mixture evenly.
Simmer the Soup:
Pour in the chicken broth, then add the shredded chicken and chopped red potatoes. Bring the mixture to a gentle boil over medium-high heat. Reduce to medium-low and simmer for 20 minutes, allowing the potatoes to become tender and the flavors to meld.
Final Touches:
Stir in the frozen peas and half-and-half, cooking for an additional 5 minutes until the peas are warmed through and the soup reaches a creamy consistency.
Prepare the Oyster Crackers (Optional):
In a large bowl, combine the oyster crackers, olive oil, thyme, kosher salt, and cayenne pepper (if desired). Mix well to coat. Spread the crackers evenly on the prepared baking sheet and bake for 10 minutes, stirring once halfway through, until golden and crisp.
Serve:
Divide the soup into bowls and garnish with the toasted oyster crackers if desired. Leftovers can be stored in an airtight container in the refrigerator for up to four days or frozen for up to two months.