Preheat the oven broiler. Melt the butter in a medium nonstick skillet over medium heat. Add the shallot and sprinkle with 1/4 teaspoon salt and a few grinds of pepper. Cook, stirring occasionally, until soft, about 5 minutes. Remove from the heat and cool. Reserve the skillet for cooking the frittata.
Meanwhile, whisk the half-and-half and eggs together in a medium bowl with 1/4 teaspoon salt and a few grinds of pepper until smooth. Add the cooled shallots, cheese, parsley and chives and stir to combine.
Heat the reserved skillet over medium heat. Pour the egg mixture into the skillet and stir a few times. Cook, undisturbed, until the bottom sets, about 5 minutes. Transfer the skillet to the oven and broil until the eggs are just set, about 5 minutes more. Remove from the oven and let stand for 5 minutes. Slide the frittata out onto a large plate or cutting board. Cut into wedges and sprinkle with more chives and serve warm or room temperature with a green salad if desired.
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