In a small bowl, soak fruit bits with water and lemon juice for several hours or overnight. Drain fruit and reserve liquid.
In a small pan, melt 1 tablespoon butter over medium-high heat. Add cornstarch, stirring quickly. Stir in syrup or honey and the water from the fruit. Stir until mixture is thick and smooth. Add fruit and simmer on low heat while making the omelet.
In a large bowl, whip together flour, eggs, half-and-half, and nutmeg until frothy. Cover bowl with a towel and place in refrigerator for 15 minutes. Remove and whip mixture once more.
Turn heat to medium high under a large frying pan. Melt 1 1/2 tablespoons butter and add egg mixture. Cover, turn heat to low, and cook for 7 to 8 minutes-until omelet is light brown on the bottom. 5. With a wooden spoon, break the omelet into bite-sized pieces and turn them over. Turn heat off and allow egg mixture to become firm on the bottom.