In a medium nonstick skillet, melt 1/2 tablespoon butter over medium heat. Add the onion and cook, stirring, until softened, about 6 minutes. Add the ham and thyme and cook over medium-high heat, stirring, for 2 minutes. Transfer to a small bowl, stir in the sour cream and cover to keep warm. Wipe out the skillet.
In a small bowl, beat the eggs with 2 tablespoons water; season with salt and pepper. In the same skillet, melt 1/2 tablespoon butter over medium-high heat. Pour in the eggs and cook, stirring, until partially scrambled. Lower the heat and tilt the pan to spread the egg evenly. Spoon the ham mixture on top and cook until the bottom is set, about 1 minute; season lightly with pepper. Fold the omelet over and cut in half.