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, stir frequently.
Add the garlic to the pot, 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, stir 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 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, 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.