Preheat oven to 450F. On a rimmed baking sheet, toss potatoes and carrots with 2 tablespoons olive oil, 1 teaspoon salt, and ½ teaspoon pepper; spread in an even layer.
Bake until browned, about 15 minutes, tossing with a spatula halfway through cooking. Set aside.
In a Dutch oven, add remaining 2 tablespoons olive oil; sprinkle bacon in an even layer.
Cook over medium heat, stirring occasionally, until crispy, 6 to 8 minutes. Remove bacon using a slotted spoon, and set aside.
Sprinkle chicken with 1 teaspoon salt and ½ teaspoon pepper; cook in batches until just browned, about 3 minutes.
Remove chicken from pan using a slotted spoon. Set aside.
Add onion, celery, and garlic to pan; cook, stirring occasionally, until softened, about 5 minutes.
Add butter; let melt. Sprinkle in flour; stir to coat vegetables, and let cook one minute, stirring constantly.
Gradually add stock, whisking constantly. Add milk, and bring to a boil.
Reduce heat to medium low; add chicken, carrots, potato, and thyme.
Let simmer until thick and creamy, about 10 minutes.
Stir in peas; taste and season with additional salt and pepper, if desired.
Sprinkle servings with bacon; serve with Thyme-Parmesan Crackers.