Fagiolini are green beans. Even in season, they don’t appear all that often on the Italian menu.
As with many vegetables, you may think of them as “overcooked”. Al dente isn’t how one thinks of one’s greens in Italy. Still, you drizzle some great olive oil on them, perhaps a squeeze of lemon, and all that negativity passes as if in a dream.
Fagiolini appear with potatoes in some versions of pasta al pesto, as in Bavette al pesto, patate e fagiolini. They might also appear in a “pie” called torta salata con i fagiolini.