Heat butter on medium heat in an 8″ non-stick pan.
In a medium bowl, beat eggs, a pinch of kosher salt, and 2 tablespoons of sugar just until whites disappear.
When butter is sizzling and foamy, but before it browns, pour in egg mixture and beat gently but continuously until eggs are very softly scrambled and creamy, 1 to 2 minutes.
Spread egg in an even layer around skillet and scrape down the edges; the top surface should be loose and creamy; if it looks liquidy or raw, cook undisturbed for another few seconds.
When the top is gently set and the bottom of the omelette is lightly caramelized, remove from heat and either roll or fold your omelette to your desired shape.
Once plated, top your omelette with a scoop of salted caramel ice cream and drizzle with a good finishing oil.