Preheat the oven to 325 degrees. Lightly spray or wipe a large baking sheet with vegetable oil.
In the large bowl of an electric mixer, combine the eggs, egg whites, and vanilla. Mix until well combined. Add the sugar, flour, soda, and salt; beat until a dough forms. Stir in the nuts.
Turn the sticky dough onto a lightly floured board. With clean floured hands and with the aid of a spatula or pastry scraper, knead the dough several times. Divide the dough in half. With floured hands, form the dough into two flattish logs, about 38 cm long and 5 cm wide. Place the logs on the baking sheet, 8 cm apart. Bake for 40 minutes. Remove and cool for 10 minutes.
Reduce the heat to 275 degrees. On a cutting board, cut the logs diagonally into 2 cm-thick slices. Arrange the slices on the baking sheet and bake for 10 to dozen minutes. Turn and bake for another 10 to dozen minutes. Turn the heat off, and leave the biscotti to crisp in the oven for 15 minutes. Remove from oven and cool completely.