Melt butter in a large heavy-bottomed pot or Dutch oven over medium heat. Add onion, thyme, and 1/2 teaspoon salt, and cook, stirring occasionally, until the onion is soft and translucent, about 8 minutes. Add rice and garlic. Cook, stirring occasionally, until fragrant, about 1 minute. Add broth, milk, and remaining 1 1/2 teaspoons of salt. Season with pepper.
Bring to a boil, stirring occasionally so that rice doesn't stick to the pot. Add chicken, stir to combine, and bring back to a boil. Reduce heat to medium-low, cover, and simmer until all liquid is absorbed and rice is tender, about 15 minutes. Add more water if the rice absorbs it all before becoming tender.
Set an oven rack about 6 inches from the heat source and preheat the oven's broiler.
Remove pot from heat. Stir in peas and 1 cup Cheddar cheese. Taste and season with more salt to taste. Sprinkle with remaining 1/2 cup Cheddar cheese.
Place under the broiler and cook until cheese turns golden brown in spots, 3 to 5 minutes. Serve immediately.
