Coarse or kosher salt and freshly ground pepperto taste
Put the broth and milk in a heavy medium-sized pot and bring to a nice simmer over medium high heat. Lower the heat to medium and slowly sprinkle the polenta in, whisking constantly. Whisk frequently until the polenta has becomes like thick oatmeal consistency, 6 to 8 minutes. Add the Parmesan, butter, salt and pepper and stir to combine thoroughly.
Grease a large plate and spoon the polenta into four circles on the baking sheet, patting them into patties about 3 to 4 inches in diameter. Refrigerate for at least 30 minutes until very stiff and cool throughout (you can cover the cakes and refrigerate for up to 2 days).
Toss the arugula with the olive oil, vinegar, salt and pepper.
Heat a very large skillet over medium high heat. Add the olive oil. Then place the four polenta cakes in the skillet and cook for 4 to 5 minutes, until the bottom is nicely browned, then flip and cook for another 4 to 5 minutes. Place a handful of arugula on each of four plates, and place a polenta cake next to the arugula. Drape a piece of salmon over each cake.
For the eggs, heat the butter in the same skillet, crack open the eggs into the skillet and cook them until they are as you like them. Place each egg on a salmon-draped polenta cake, season with salt and pepper and serve hot.