Preheat the oven on the grill function to 200°C (180°C fan). Soften 4Tbsp butter in a small bowl. Peel, halve and finely dice the onions. Peel and mince the garlic. Add the tomatoes and their juices to a bowl and crush them with your hands, to make a chunky sauce. Strain and rinse the beans. Pluck the basil leaves from the stems. Finely chop the basil stems.
In a large oven-safe pan on medium heat, sauté the onions and ½ of the garlic with 4Tbsp olive oil on medium heat for ca. 2min. Then add the tomatoes and the oregano sprigs. Cover with a lid and bring to a simmer. Once simmering, stir in the beans, 100ml water and a large pinch of salt and pepper. Cover again and simmer for ca. 10min, until the sauce thickens around the beans.
Remove the pan from the heat and discard the oregano sprigs. Add more water as needed to keep the beans saucy and season to taste with salt and pepper. Tear the mozzarella into pieces and add on top of the beans. Put the pan in the center of the oven and grill for 10min., until the sauce is stewy and the cheese has melted and slightly browned.
Mix the chopped basil stems into the softened butter, along with ½ of the remaining garlic and a large pinch of salt. Cut the breads into 2cm slices, but don't cut all the way through. Spread the butter in between the slices. After the beans grilled for 10min., turn the oven from grill to bake. Put the breads on a baking tray and add it to the oven with the beans. Bake both for 5min.
In a tall container, blend 2Tbsp mayonnaise, 2Tbsp olive oil, 1Tbsp of vinegar, 2tsp mustard, ½ of the shaved cheese, the remaining garlic and a large pinch of salt and pepper until smooth.
Sprinkle the remaining shaved cheese over the beans. Garnish the beans with the basil leaves and a drizzle of olive oil. Toss the salad with enough dressing to coat the leaves. Season to taste with salt. Serve the beans in wide bowls, topped with the salad. Serve the bread on the side, using it to scoop up any sauce from the beans.