Heat the olive oil in a large pot over medium heat. Add the diced onion and cook until softened, about 5 minutes.
Stir in the minced garlic and cook for 1 minute until fragrant.
Add the white beans, crushed tomatoes, vegetable broth, and oregano. Season with salt and pepper.
Bring the mixture to a simmer and cook for 15 minutes, stirring occasionally.
Use an immersion blender to puree the soup until mostly smooth, leaving a few beans for texture. Alternatively, transfer half the soup to a blender, puree, and return it to the pot.
While the soup simmers, heat a skillet over medium heat. Butter one side of each bread slice.
Place 4 bread slices butter-side down in the skillet. Top each with 2 cheese slices and another bread slice, butter-side up.
Cook the sandwiches until golden on both sides and the cheese melts, about 3 minutes per side. Slice and serve with the soup.