Sauté the Vegetables: In a large soup pot, heat the butter and olive oil over medium-high heat. Once melted and shimmering, add the diced celery, carrots, and onion. Cook for about 5 to 7 minutes, stirring occasionally, until the vegetables soften and start to release their aroma.
Build the Flavor Base: Add the minced garlic to the softened vegetables, stirring constantly for about 30 seconds to avoid burning. Sprinkle the flour evenly over the mixture and continue to cook for another minute, stirring to create a light roux that will help thicken the soup.
Add Broth and Seasonings: Slowly pour in the chicken broth while stirring to dissolve the flour completely. Mix in the Italian herbs, then carefully place the raw chicken breasts into the pot. Increase heat and bring the mixture to a gentle boil.
Simmer the Soup: Reduce the heat to low, partially cover the pot with a lid left slightly ajar, and let the soup simmer for 15 minutes. This allows the chicken to cook through while infusing the broth with flavor.
Cook the Orzo: Remove the lid and add the orzo pasta to the simmering soup. Keep the heat at medium-low and cook uncovered, stirring frequently to prevent the pasta from sticking to the bottom, until the orzo is tender—about 10 minutes.
Shred and Return Chicken: Using tongs or a slotted spoon, remove the chicken breasts from the soup and place them on a cutting board. Slice or shred the chicken into bite-sized pieces, then return it to the pot.
Final Flavor Touches: Stir in the fresh lemon juice and chopped parsley. Taste the soup and adjust seasoning with salt and pepper as desired. Serve hot immediately for the best flavor and texture.