Fresh corn polenta is made with grated fresh corns. The texture is identical to polenta but tastes so much more flavorful.
Put two pieces of parchment paper on a baking sheet. Shuck the corn and remove any silk.
Snap each ear in half. Use the largest holes of a box grater, grate the corn to the cob over the parchment to capture all of the kernels and corn juices. Carefully lift the paper and pour everything into a bowl.
Over medium-low heat, in a saucepan, add the butter. Add the corn and its juices, the salt, and the pepper and cook, stirring constantly, about 10 minutes.
Don’t add any liquid unless you really have to, but if the corn sticks and looks very dry, add 1 tablespoon of water. Stir and taste often until no liquid is left and none of the corn tastes raw.
Turn off the heat and stir in the mascarpone. Divide the corn among four plates.
Heat nonstick sauté pan over medium-high heat. Add olive oil and heat until the surface is rippling slightly but not smoking.
Crack the egg into the hot pool of oil. If the oil is hot enough, the white should spread only a little. Use a spoon to quickly baste the hot oil onto the egg white, being careful to avoid the yolk. Continue spooning the hot oil onto the white until the white is almost set, about 45 seconds.
When the white is opaque, use a spatula to transfer the egg directly on top of the polenta.
Repeat with the remaining eggs, adding more olive oil as necessary, then garnish each plate with chives, additional salt and pepper to taste, and serve immediately.