In a 3- to 4-quart pan, whisk 7 cups water with the cornmeal, baking soda, and salt until well blended. Bring the mixture to a boil over high heat, stirring. Reduce heat and simmer, uncovered, about 10 minutes, stirring occasionally.
Mix 1 cup water with cornstarch until smooth. Whisk into cornmeal mixture. Turn heat to high and stir until boiling again; stir 1 minute longer. Remove from heat.
Place a nonstick frying pan, 9 1/2 to 10 inches across the bottom, over medium heat.
When pan is hot, rub lightly with lard, then pour in 1/4 cup of the cornmeal mixture. Tilt pan, swirling batter to coat bottom. Cook until batter stops bubbling, begins to curl away from pan sides and is dry, 5 to 7 minutes. Slide a wide spatula under sheet of paper bread (the first sheet may be hard to handle), lift out, and lay flat (if bread breaks, lift out and save pieces).
Repeat step 4 with more batter, but when edges of bread begin to curl from pan sides and bread is still bubbling a little in the center, lay the first piece, along with any broken pieces, on top of the piece in the pan. Cook until bread feels dry in the center when touched, pressing flat if it puffs up, 5 to 7 minutes. Lift bread out with spatula and fold up opposite sides, overlapping to make a loose roll (at left). (If bread is not cooked dry, it loses its crispness when it cools; if this happens, unroll, return to pan, and toast again until dry.) Set roll on a rack.
Repeat steps 4 and 5 with remaining batter to make each double sheet of paper bread. Serve immediately, or store airtight at room temperature up to 1 week.