In a bowl, whisk eggs, 1/4 cup water, and salt until blended.
Set a 10- to 12-inch nonstick frying pan over high heat. Add oil, ham, chiles, onion, and tomato; stir often until onion is limp, about 5 minutes. Reduce heat to medium-high. Add egg mixture, then sprinkle evenly with cheese. When a thin layer of egg is set on pan bottom, in about 30 seconds, push cooked egg toward one side of pan with a wide spatula, letting uncooked egg flow underneath. Continue cooking and pushing egg mixture until eggs are softly set but still moist, 3 to 4 minutes total. Stir to turn eggs over. Remove pan from heat.
Wrap tortillas in plastic wrap and heat in a microwave oven on full power (100%) just until hot and steamy, 15 to 30 seconds. Lay tortillas, one at a time, on a board and spoon about a fourth of the egg mixture across each, just below center. Wrap tortillas tightly around filling. If desired, cut each burrito in half diagonally; stack on plates. Offer salsa, salt, and pepper to add to taste.