In a large heavy-bottomed pot over medium heat, add olive oil and butter. When butter melts and foams, add the diced onion, carrots and celery. Cook, stirring occasionally, until softened and translucent, about 6–8 minutes. Add the minced garlic and cook 30–45 seconds until fragrant.

Make a roux

Sprinkle the flour over the softened vegetables and stir to coat. Cook the flour with the vegetables for 1–2 minutes to eliminate the raw taste, stirring constantly so it doesn't brown.

Add liquid and aromatics

Slowly whisk in the chicken (or vegetable) broth until smooth. Add the bay leaf and thyme sprigs. Increase heat to bring the mixture to a gentle boil, then reduce to a simmer.

Simmer with potatoes

Add the cubed potatoes to the simmering broth. Simmer, partially covered, until potatoes are almost tender, about 12–15 minutes depending on size.

Finish with milk

Lower heat to gentle; stir in the milk or half-and-half. Do not boil vigorously after adding milk—just heat through for 3–4 minutes. Season with 1 tsp kosher salt and 1/2 tsp black pepper now; adjust later after adding other ingredients.

Vegetables & Add-ins

Add the green beans to the pot and simmer until they are tender-crisp, about 4–5 minutes. Stir in the corn and peas and cook 2–3 more minutes until heated through. If using cannellini beans, add them now and warm through for 2–3 minutes. Taste the broth and adjust salt to about 1.5 tsp total and black pepper to 1 tsp, adding more if desired.

Finishing & Garnish

Remove and discard the bay leaf and thyme sprigs. Stir in the lemon juice, chopped parsley and chives. If using, stir in the grated Parmesan until melted and incorporated for a richer flavor. Taste and adjust seasoning with additional salt, pepper, or a squeeze more lemon if desired.

Ladle the soup into bowls and finish each serving with a grind of cracked black pepper and a sprinkle of extra parsley or chives. Serve hot with crusty bread.