Heat olive oil and butter in a large Dutch oven or heavy-bottomed pot over medium heat. Add the finely diced onion and cook until softened and translucent, about 5-7 minutes. Add the minced garlic and cook for another 1-2 minutes until fragrant, being careful not to brown it.

Sprinkle the all-purpose flour over the onion and garlic mixture. Whisk constantly for 1-2 minutes, allowing the flour to cook out and form a light roux. This will thicken the soup.

Gradually whisk in the chicken broth, a little at a time, to prevent lumps. Once all the broth is incorporated and smooth, stir in the whole milk and heavy cream.

Bring the soup to a gentle simmer, stirring occasionally. Let it simmer for 10-15 minutes, or until the soup has thickened slightly to your desired consistency.

Reduce the heat to low. Stir in 1 cup of freshly grated Parmesan cheese until it is fully melted and incorporated. Do not boil the soup after adding the cheese, as it can cause the cheese to separate.

Season the soup with salt, black pepper, and a pinch of nutmeg (if using). Taste and adjust seasonings as needed.

For a smoother soup (optional), carefully use an immersion blender to blend a portion of the soup directly in the pot, or transfer 1-2 cups to a regular blender (blend in batches, being careful with hot liquids, and vent the lid) and return to the pot.

Ladle the hot soup into bowls. Garnish with extra grated Parmesan cheese and fresh chopped parsley. Serve immediately.