Step 1

Arrange a rack in the middle of the oven and heat the oven to 375°F.

Step 2

Dice 1 medium yellow onion (about 1 1/2 cups). Peel and halve 3 medium carrots and halve 3 medium celery stalks lengthwise. Cut both crosswise into 1/2-inch wide pieces (about 1 cup each). Mince 2 garlic cloves. Pick the leaves from 1/2 medium bunch fresh thyme and coarsely chop (about 2 teaspoons), or measure out 1 teaspoon dried thyme.

Step 3

Melt 4 tablespoons of the unsalted butter in a large skillet over medium-high heat. Add the onion, carrots, celery, thyme, 1 1/2 teaspoons kosher salt, and 1/2 teaspoon black pepper. Sauté until the vegetables begin to soften and brown, 5 to 7 minutes.

Step 4

Add 1/4 cup all-purpose flour and the garlic. Stir to coat the vegetables in the flour and cook until fragrant and no dry spots remain, 30 seconds to 1 minute. Slowly pour in 2 cups low-sodium chicken broth and 1 cup whole or 2% milk, scraping the bottom of the skillet to release any browned bits. Bring to a boil.

Step 5

Reduce the heat to maintain a simmer and cook until thickened enough to coat the back of a spoon, 4 to 5 minutes. Meanwhile, shred or dice 1 pound cooked chicken into bite-sized pieces (about 4 cups).

Step 6

Add the chicken and 1 cup frozen peas to the skillet, and stir to combine. Taste and season with more kosher salt or black pepper, as needed. Transfer to a 9x13-inch baking dish and spread into an even layer. Open 1 (about 16-ounce) can refrigerated biscuits, separate the biscuits, and arrange in a single layer over the filling.

Step 7

Bake until the biscuits are golden-brown and cooked through, and the filling in the center of the baking dish is bubbling, 20 to 30 minutes. Check after 15 minutes, rotate the baking dish, and cover with aluminum foil if the biscuits are browning too quickly.

Step 8

Melt the remaining 2 tablespoons unsalted butter in the microwave or on the stovetop. Brush the butter over the tops of the biscuits.