Melt the butter in a large pot over medium heat. When the butter has melted and starts to foam, add the onions, carrots and celery.
Cook the vegetables gently for 5 minutes until the onions become translucent and start to soften, stirring frequently.
Add the garlic to the pot and cook for 1 minute, stirring frequently.
Add the flour while stirring constantly until well incorporated. Cook the mixture for 2–3 minutes, stirring almost constantly to cook out the flour taste a bit.
Slowly add in the chicken broth while stirring until it is incorporated. Use a whisk to make it easier to dissolve the flour. If you don't have one, stir vigorously with a wooden spoon.
Add the thyme bundle to the soup.
Raise the heat to medium-high and bring the soup to a gentle simmer (it will take about 3–5 minutes).
Lower the heat to medium-low, half-cover the pot and continue simmering for 20 minutes, stirring occasionally. Keep the soup at a simmer. If it starts boiling too vigorously, lower the heat a bit.
Use tongs to remove the thyme bundle.
Add the cooked chicken and wild rice. Raise the heat to medium-high and bring to a simmer. Lower the heat to medium-low, half-cover and cook for 5 minutes, stirring occasionally to make sure the rice is not sticking to the bottom.
Take the pot off the heat and add the half-and-half. Stir to combine.
Add the black pepper and stir.
Return the pot to low heat and warm through. Do not let it come to a boil. Taste the chicken and wild rice soup and add salt, if needed. As a reference, we did not add extra to ours.