In a large pot, melt 2 tablespoons of unsalted butter and 1 tablespoon of olive oil over medium heat. Add 1 diced yellow onion, 2 chopped carrots, and 2 chopped celery stalks, sautéing for about 6–8 minutes until they are soft and fragrant. Stir in 3 minced garlic cloves and cook for an additional minute until the garlic becomes aromatic.

Sprinkle in 1/4 cup of all-purpose flour over the sautéed vegetables, stirring consistently to combine. Continue cooking for 2–3 minutes until the mixture forms a paste and becomes slightly golden.

Gradually pour in 4 cups of chicken broth while stirring vigorously to prevent lumps from forming. Bring the mixture to a gentle simmer, cooking uncovered for 10–12 minutes. If adding a small diced potato, toss it in at this stage.

Lower the heat to medium-low and gently stir in 1 cup of whole milk and 1/2 cup of heavy cream. Gradually add 1 cup of grated aged Parmesan cheese, stirring continuously until it melts completely.

Mix in 1 teaspoon of dried thyme, 1 teaspoon of dried basil, and a pinch of crushed red pepper flakes. Season with salt and black pepper to taste, simmering for an additional 5–10 minutes.

If you prefer a smooth consistency, use an immersion blender to puree the soup until it reaches your desired texture. For a rustic approach, leave it slightly chunky.

Before serving, taste your soup and adjust seasoning as needed. Add a splash of lemon juice or more herbs for enhancement if required.