In large saucepan, cover 6 eggs with at least 1 in. of cold water. Cook over high heat until water boils. Remove from heat; cover and let stand 9 minutes. Drain and rinse with cold water to stop cooking. Refrigerate until completely cool. Peel.
In large bowl, combine flour, cornmeal, baking powder, baking soda, salt, mustard, black and cayenne peppers. In another bowl, whisk buttermilk, oil, and egg until well combined. Add wet ingredients to dry ingredients and stir until nearly combined. Fold in cheese and chives.
Spoon about 1/4 cup batter into each muffin cavity. Place peeled egg on top. Divide remaining batter evenly, using fingers to mold the batter to the top of the egg if necessary.
Bake 22-24 minutes or until golden brown. Serve warm.