In a 12 to 14-inch saute pan, heat 2 tablespoons over medium-high heat until hot, but not smoking. Add the onion and anchovies and cook until the anchovies are breaking up, about 5 minutes. Add 1 thinly sliced garlic and continue cooking until the anchovies have dissolved completely, about more 5 minutes.
Meanwhile, remove the skin from the reconstituted codfish, and slice the flesh in half lengthwise. Cut both pieces of fish cross-sectionally into 1-inch pieces.
Place the flour and 2 tablespoons of Parmigiano into separate shallow bowls. Dredge a piece of fish in the flour, then the cheese and set aside. Repeat with the remaining pieces of fish. In a large skillet, heat 2 tablespoons extra-virgin olive oil. Add the fish pieces and the remaining 10 thinly sliced garlic cloves. Cook the fish until golden, then flip. Add the onion anchovy mixture and the milk and bring to a boil. Flake the fish with a fork to see if it is cooked. Add 1/2 cup grated cheese and a drizzle of extra-virgin olive oil. Divide the hot polenta evenly among 4 warmed bowls. Spoon the hot codfish and sauce over the polenta and serve immediately.
In a 3-quart saucepan, heat water and salt until boiling. Drizzle in polenta in a thin stream, whisking constantly, until all polenta is incorporated and mixture begins to thicken. Switch to a wooden spoon, pull off heat and continue to stir until thick as paste.