Dice the carrots, celery, onion into 1/2-inch pieces, and cut zucchini into half-moons. Mince the garlic and cut chicken breasts into 1-inch chunks. Melt butter in a large pot over medium heat, then add the carrots, celery, onion, zucchini, and garlic. Stir occasionally and cook covered for 30 minutes until the vegetables are softened. I like to use Kerrygold unsalted butter here—it adds a subtle richness that really elevates the broth's flavor.

Add the chicken broth, chicken chunks, salt, and pepper to the pot with the softened vegetables. Bring to a boil over medium-high heat, then reduce to a simmer and cook for 8 minutes until the chicken is just cooked through. The chicken will continue to cook slightly from residual heat, so don't overcook it at this stage—you want it tender when the lemon egg mixture is added.

While the chicken simmers, whisk the room-temperature egg yolks with lemon juice in a separate bowl until well combined. Once the chicken is cooked, slowly ladle about 1/2 cup of hot broth into the egg mixture while whisking constantly—this gradually raises the temperature of the eggs so they won't scramble when added to the soup. I always temper the eggs this way rather than dumping them directly in; it's the key to getting a silky, creamy texture instead of scrambled bits.

Remove the pot from direct heat or reduce to the lowest simmer. Slowly pour the tempered egg mixture into the soup while stirring gently and constantly to distribute it evenly throughout. Stir in the fresh parsley and dill, season to taste if needed, and serve immediately. The gentle heat will create a silky, velvety soup without any risk of the eggs curdling.