Heat the oven to 400 F / 200 C. Slice the top quarter off each garlic head to expose the cloves, then set both heads on a small sheet of foil, cut side up.

Drizzle the exposed cloves with 1 tablespoon olive oil, wrap tightly in the foil, and roast for 40 to 45 minutes until the cloves are soft, golden, and easily pierced with a fork.

Unwrap and let the garlic cool for 10 minutes, then squeeze the cloves out of their skins into a small bowl and mash lightly with a fork.

While the garlic roasts, heat the remaining 3 tablespoons olive oil in a 5-quart / 4.7 L Dutch oven over medium heat. Add the onion, celery, and carrot, and cook for 7 to 8 minutes, stirring occasionally, until softened.

Add the mashed roasted garlic to the pot and stir for 1 minute until fragrant.

Add the cannellini beans, vegetable broth, bay leaf, thyme, salt, and pepper. Bring to a simmer and cook for 20 minutes, stirring occasionally, until the vegetables are fully tender and the broth tastes rounded, not raw.

Remove the bay leaf. Use an immersion blender to puree about half the soup directly in the pot, leaving the rest chunky, so the finished soup is thick but still has whole beans and bits of vegetable. Stir in the lemon juice and taste for salt.

Heat the 2 tablespoons olive oil in an 8-inch / 20 cm skillet over medium heat. Fry the sage leaves in a single layer, without crowding, for 30 to 45 seconds per side until crisp and darkened at the edges. Transfer to a paper towel.

Ladle the soup into bowls and top each with crispy sage leaves and a scattering of grated parmesan, if using. Serve with crusty bread.