Whisk eggs, cream and chives in a bowl until combined, season to taste and set aside.
Split lobster lengthways, remove tail meat. Remove intestinal tract and discard, coarsely chop meat and set aside (you’ll need 150gm). Preheat oven to 180C. Melt butter and oil in a 25cm-diameter ovenproof frying pan, add meat and sauté over medium heat until opaque (1-2 minutes). Transfer to a warm plate, return pan to heat.
Pour egg mixture into pan and cook until just starting to set (20-30 seconds). Draw cooked egg into centre of pan with a wooden spoon or spatula, tipping pan to allow uncooked mixture to flow into spaces. Transfer to oven and bake until top is almost set (1-2 minutes), then place lobster meat, crème fraîche and watercress along centre of omelette and roll to enclose. Cut into four pieces and serve immediately.