Heat stock and water in a medium-size pot over moderate heat.
In a large, deep skillet, heat EVOO, 2 turns of the pan, over medium-high heat. Melt butter into the oil; when it foams, add the pasta and stir. Once golden, add vegetables, garlic and herbs. Stir 2-3 minutes to soften then add tomato paste, stir to incorporate then add wine. Stir until wine is absorbed then add 2 cups of the warm stock and water. Stir gently until absorbed. Continuing adding the warm stock and water in stages until pasta is almost al dente and starchy/creamy. Add the white beans, let them heat through a minute then stir in cheese.
Serve in shallow bowls, drizzled with more EVOO and pass more cheese at the table.