To make the crepes: Using a fork, stir enough water, about 1/3 cup, into the flour to form a thick paste. Continue mixing with the fork until all clumps have disappeared and the blend is homogenous. Mix in salt to taste.
Heat a 6-inch, non-stick omelet pan over high heat. Grease lightly with a little extra-virgin olive oil, and add a small ladleful (2 tablespoons) of the crepe batter to the hot pan. Immediately swirl the pan to coat bottom with batter, and cook until crepe loosens from side of skillet and underside is pale golden, about 3 minutes. Ideally, you want the crepe to be an even color on both sides. Remove the crepe from the pan and leave on a plate, covered with waxed paper. Repeat with the remaining batter, re-greasing the pan when needed, until all the batter has been used, adding each new crepe to the pile beneath the paper towel. Set the crepes aside to cool.
In the bowl of a food processor, combine the basil, pine nuts, garlic, and 2 teaspoons pecorino. Pulse gently, just until roughly chopped. Pour in the olive oil bit by bit as you continue to pulse the basil and cheese mixture. When enough oil is in the bowl to make the ingredients pulse smoothly, pulse continually until all the oil has been added and the mixture is smooth and homogenous. Season, to taste, with salt. Transfer to a large bowl.
Bring 6 quarts water to a rolling boil and add 2 tablespoons salt.
When the crepes are completely cool, use a sharp knife to slice each crepe into thin fettucine-like strips, about 1/2-inch each. Drop the crepe into the boiling water just like you would dried pasta. Remove after 45 seconds. Drain well.
Divide the "pasta" equally among 4 serving bowls. Top with pesto as desired and the remaining pecorino. Serve immediately.