Crack eggs into medium mixing bowl. Add half and half and sea salt, then aggressively whisk ingredients until mixture is uniform and completely combined, without any streaks or separation. For really creamy eggs, let mixture sit for 5 minutes, up to 15 minutes, to give salt time to fully absorb.
Warm nonstick pan over medium heat and add cold butter. Allow butter to melt and fully coat pan. Once butter starts to foam, pour egg mixture into center of pan and immediately reduce heat to medium-low.
Let egg mixture sit in pan until edges begin to barely set, then use spatula to gently push eggs from one side of pan to other.
Continue process of pushing eggs around pan, pausing after each swipe to allow uncooked egg to settle on pan and cook.
Once eggs are roughly 50% cooked, add shredded cheddar and truffle oil. Stir well, using broad strokes, until oil is integrated and cheese is melted.
Remove pan from heat when eggs are cooked but still shimmery from moisture. Top with freshly cracked black pepper and chopped chives. Serve immediately.