Lightly whisk eggs in a bowl and season to taste with salt and freshly ground black pepper. Heat a frying pan (about 22cm diameter) over medium-high heat, add 10gm butter and swirl to coat pan, then add eggs. Use a spatula to push egg towards the centre of the pan, tilting the pan to allow the uncooked egg to run underneath until the egg is no longer runny and the texture is light and layered, then leave and cook until egg is set (1-2 minutes). Knock pan to loosen omelette, then carefully roll and fold onto a plate.
Heat a small saucepan over high heat, add remaining butter and heat until foaming (2-3 minutes). Add miso, stir to combine, then add crabmeat and stir gently until just combined and heated through (1 minute). Stir in chives and lemon juice to taste, then spoon onto omelette. Serve hot.
Note Shiro (white) miso paste is available from health food shops, select supermarkets and Japanese and Asian grocers.