1. Peel the onions and place them in a saucepan with the butter, salt, and pepper to taste and 2 tablespoons of chicken stock.
2. Cover and simmer for 20 minutes.
3. In a large casserole, sauté the 4 pigeons with the olive oil until they are golden all over.
4. Add the bay leaf, the Grappa, the onions, and the rest of the chicken stock.
5. Cover and simmer on low for 15 minutes.
6. Add the chopped carrots and cook for 10 more minutes. In the meantime, cook the borlotti beans in salted boiling water for 5 minutes.
7. Drain the beans and put them on a large serving plate.
8. Top with the pigeons, carrots, and onions and serve immediately.