Bring 6 cups of water to a boil in a large heavy saucepan, add 2 teaspoon salt. Gradually whisk in the cornmeal. Reduce heat to low and cook until the mixture thickens and the cornmeal is tender, about 15 minutes, stirring often.
Turn off the heat and add the butter, stirring until the butter is melted. Stir in chopped green onions. Pour mixture into a buttered 11 x 7 baking dish. Cool. Refrigerate until set. Cut polenta in large squares or triangles. You can also cut the polenta with cookie cutters for a more festive look.
Make Grilled Polenta
Brush both sides with olive oil.
Heat a large skillet over medium heat. Lay polenta slices in skillet and cook until nicely browned and crisp, about 10 minutes. Flip over and brown the other side. Set aside and keep warm if necessary.
Make Sunny Side Up Eggs
Crack each egg into a custard cup.
In a nonstick skillet over medium-low heat, melt butter.
When butter has melted and begins to bubble, gently pour the eggs into the skillet, one at a time. Cook until the whites are opaque and the yolk has set, about 2 to 3 minutes, depending how you like your eggs.
Sprinkle with salt and pepper to taste.
Serve on top of grilled polenta.