For the Soup: In a large Dutch oven or soup pot, heat oil over medium heat until warmed. Add onions and a pinch of salt and cook, stirring often, until softened but not browned, about 4 minutes. Add beans along with enough water to cover (about 1 quart/liter) and a large pinch of salt and bring to a simmer over high heat. Lower heat to maintain simmer and cook until beans are al dente (almost tender but still retain a slight bite), about 40 minutes (though time will depend heavily on the type of beans you use so check doneness and prioritize that over the time estimate).

Add string (or Romano) beans, zucchini, and carrots, along with enough additional water to cover; season with a generous pinch of salt. Bring to a simmer over high heat, then reduce heat to maintain simmer and cook until vegetables are very tender, about 1 hour.

Stir in tomato and potatoes and cook until potatoes are very tender, about 30 minutes. Top up with additional hot water as needed if the soup becomes too dry: The final consistency of the soup will depend on your preference, some people like it brothier, others thicker like a hearty stew. Season with salt.

For the Pistou: If using a mortar and pestle, crush garlic with a pinch of salt until sticky and pasty, and no chunks remain. Next, pound in basil, a handful at a time and adding salt as you go, making sure leaves are well pounded and no longer stringy or in large bits before adding the next handful; salt acts as an abrasive and helps break down the basil, but be careful not to add too much and make the pistou too salty. Pound in the olive oil, then stir in the minced tomato. If using a blender or food processor, process garlic with the basil, olive oil, and salt until a smooth sauce forms. Stir in minced tomatoes.

To Serve: If using pasta, stir the cooked pasta into the soup, adjusting consistency with additional hot water if it's too thick; season to taste with additional salt as needed. Ladle soup into serving bowls, passing the pistou and cheese alongside for diners to stir into their soups to taste.