Heat the olive oil in a large pot or Dutch oven over medium heat. Once the oil is hot, add the diced onion, carrots, and celery. Sauté for about 5 minutes or until the vegetables begin to soften and the onion becomes translucent. Add the minced garlic and cook for another 1-2 minutes until fragrant.

Add the chicken breasts (or thighs) to the pot. Cook for about 6-7 minutes per side until the chicken is fully cooked and browned. Once the chicken is done, remove it from the pot and set it aside to cool slightly.

In the same pot, add the diced potatoes and chicken broth. Stir everything together, scraping the bottom of the pot to release any browned bits. Bring the mixture to a boil, then reduce the heat to a simmer. Let it cook for about 15-20 minutes, or until the potatoes are tender when pierced with a fork.

While the potatoes are cooking, shred the cooked chicken using two forks. Once the potatoes are tender, add the shredded chicken back into the pot.

Pour in the heavy cream (or milk) and stir well to combine. Let the soup simmer for another 5-10 minutes, allowing the flavors to meld together. If the soup is too thick, you can add a little more broth or water to reach your desired consistency.

Taste the soup and season with salt and pepper as needed. If you like, toss in some fresh herbs like thyme or parsley for an extra burst of flavor.