Place the butter in a saucepan and melt it gently over low heat, taking care not to let it brown.
Add the sliced leek, minced garlic, thyme sprig, and rosemary to the pan. Cook slowly for 8 to 10 minutes, stirring occasionally, until the leeks become soft, aromatic, and delicate in texture without taking on color.
Pour in the white wine and allow it to simmer for a couple of minutes, loosening the flavorful bits from the base of the pan and deepening the overall taste.
Add the rinsed cannellini beans, vegetable broth, salt, and black pepper. Stir well to combine the ingredients evenly.
Fold in the grated cheddar and let it melt into the mixture, creating a creamy, cohesive sauce. Continue cooking for another 8 to 10 minutes, until the beans are fully warmed through and the flavors have blended.
Finish the dish by grating the vegetarian pecorino over the beans. Add the lemon juice and zest, then adjust the seasoning as needed.
Sprinkle the chopped parsley over the top just before serving for freshness and color.
Serve immediately with crisp slices of bread alongside, allowing the bread to complement the rich, cheesy sauce.