Warm the oil in a large soup pot over medium heat. Add the onion, carrots, and celery. Cook until the vegetables are tender, about 6 minutes. Add the broth, chicken, and potatoes; cover; and bring to a gentle simmer over high heat. Uncover, reduce the heat to low, and poach at a bare simmer until the chicken is no longer pink in the center (160ºF internal temperature), about 15 minutes.
Remove the chicken to a cutting board to cool.
Meanwhile, in a small heatproof bowl, use a fork to stir together the butter and flour into a smooth paste. Gradually stir the paste, a few bits at a time, into the stew and simmer over low heat until the stew thickens, 3 to 4 minutes.
In the same bowl, whisk together the egg yolks and half-and-half. Whisk ¼ cup of the hot stew into the eggs; then whisk the egg mixture back into the stew. Cook over low heat for 1 minute.
Cut the chicken into bite-size cubes and stir them into the stew along with the peas, salt, and pepper. Heat through; serve.
Tip: For more flavor, stir in ½ teaspoon of dried thyme along with the broth. You can also add a tablespoon of chopped fresh parsley just before serving.
The stew can be made up to 3 days ahead of time and refrigerated in an airtight container. Reheat it in the microwave or a saucepan just before serving.