To a wide/shallow Dutch oven, or a large skillet (enough so 4 chicken quarters can lay flat), melt butter and oil. Once hot, add chicken and garlic. Sear chicken 3 minutes on each side.
Add enough broth so liquid level comes halfway up chicken (2-3 cups). Add lemon juice, zest, and seasonings.
Bring to a boil, reduce and simmer covered for 30-40 minutes, or until fully cooked. Remove chicken and pull meat off bones, cutting into bite-sized pieces. Set aside. Strain cooking liquid and reserve.
While chicken is cooking, sauté a mix of yellow, orange and red peppers, until nice and golden brown in places. Set aside.
To a large stock pot, add reserved cooking liquid from chicken, chicken pieces, peppers, along with another 2-3 cups of water. Bring to a boil.
If desired, slowly drizzle in 2 eggs yolks while whisking continuously - this will help thicken soup as well as add a rich creaminess to soup.
Reduce heat to a simmer and stir in cream (or canned coconut milk).
Just before serving, stir in 1-2 Tbsp of Italian Walnut Pesto (to taste), garnish with parsley.