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Over medium-high heat, melt butter in a medium-size skillet, preferably well-seasoned cast iron or nonstick.
Beat eggs with salt, pepper and, if using, Parmesan cheese. Pour eggs into pan, swirl and turn heat to low. Using a heat-proof rubber spatula, stir eggs constantly until very loosely set and slightly runnier than you like them.
Remove pan from heat and drizzle pesto on eggs. Off heat, give eggs one more gentle stir -- enough to finish cooking them and to distribute pesto in dark green streaks. Scatter ricotta on eggs and drizzle with more pesto if desired. Serve at once.