Rinse tomato, core, cut in half crosswise, and squeeze out seeds. Chop tomato.
In an 8- to 10-inch frying pan over medium-high heat, combine oil and shallots. Stir often until shallots are limp, 3 to 4 minutes. Add tomato, 2 tablespoons chives, and basil; stir until tomato juices evaporate, 3 to 5 minutes. Add salt and pepper to taste. Keep warm.
In a bowl, beat eggs to blend with 1 tablespoon water and 1/8 teaspoon salt. Also, break goat cheese into small pieces.
In a 6- to 7-inch nonstick frying pan over medium-high heat, melt 1 teaspoon butter. When sizzling, add 1/2 the egg mixture. As eggs firm on the bottom, lift with a spatula to let uncooked portion flow beneath. Shake pan often to keep eggs freely moving. When eggs are softly set, in 1 to 2 minutes, spoon half the tomato mixture and chevre down center of omelet, in line with pan handle.
To fold, tilt pan at about a 45° angle parallel to filling. With spatula, fold about 1/3 of upper side of omelet down over the filling. Hold pan over plate and shake so unfolded edge slips out onto it, then quickly flip pan over to fold omelet out onto dish. Keep warm and repeat to cook remaining omelet. Sprinkle omelets with remaining chives. Add salt and pepper to taste.