To make the dough, mound the flour in the center of the work surface. Make a well in the center of the flour. Add the 3 beaten eggs and 1/4 cup water to the center of the well and begin to incorporate the liquid into the flour with a fork, a little at a time, until all liquid is incorporated. Work the dough until it comes together in 1 mass. Knead the dough for 10 minutes, then allow it to rest for 30 minutes.
Bring 6 quarts of water to a boil and add 2 tablespoons salt. Cook the cabbage in the boiling water until it is tender, about 20 minutes. Drain the cabbage and refresh in an ice bath. Once the cabbage is cooled, drain again and press between clean kitchen towels or paper towels to absorb all excess moisture. Finely chop the blanched cabbage and set aside.
In a 14 to 16-inch skillet, heat the butter until it foams and subsides. Add the salami and cook over high heat 5 minutes, until the meat is well-browned. Add the cabbage and cook over medium heat, stirring to keep the meat or cabbage from sticking, until the cabbage is of an almost creamy consistency. Remove from heat and allow to cool.
Once the mixture is cooled, stir in the grated cheese and the remaining 1 egg. Season with salt and pepper and refrigerate, covered, until ready to stuff the pastry.
Roll the dough out to the thinnest setting on a pasta rolling machine. Cut the dough into 3-inch squares. On half of the squares, place a tablespoon of the cabbage and salami mixture. Cover each filled square with another square, brush with water and press around the edges to seal.
Bring 6 quarts of water to a boil and add 2 tablespoons salt. Cook the pastries in the boiling water for 5 minutes. Remove from the water, drain and serve immediately.