Heat your Dutch oven or large pot over medium-high heat and add the olive oil. Once it shimmers, add the chicken pieces and sear them for 5-6 minutes until they get golden on the outside—you're not cooking them through yet, just creating color. Work in batches if your pot feels crowded so you don't steam them. Transfer the chicken to a plate and set aside.
In the same pot, add your diced onion and cook for 3 minutes until it softens and smells amazing. Add the minced garlic and let it warm through for about 30 seconds—don't let it burn or it'll taste bitter. Stir in the carrots and celery, cooking for another 2 minutes so they release their flavors into the oil.
Pour in your chicken broth and use a wooden spoon to scrape up any browned bits stuck to the bottom of the pot. These stuck bits are literally flavor gold (seriously, don't skip this step). Add back the chicken pieces, then drop in the bay leaf and sprinkle the dried thyme over everything.
Bring the whole thing to a boil, then reduce heat to medium-low and simmer for 20 minutes without a lid. The chicken should be mostly cooked through by now, and the broth will smell incredible. This is when I taste and adjust my salt and pepper because every broth is a little different.
Add the cubed potatoes now and let everything cook together for another 15 minutes. You'll know they're done when you can pierce them easily with a fork. The spring chicken stew recipe should be bubbling gently at this point, not aggressively boiling.
Stir in the frozen peas and cook for 5 more minutes. Don't panic if the peas look a little pale at first—they brighten up as they warm through. Taste again and adjust your seasonings because this is your last chance before serving.
Remove from heat and stir in the shredded cheddar cheese until it melts completely into the broth. This makes the whole thing rich and comforting without being heavy. Ladle into bowls and enjoy while it's steaming hot.