Combine white wine, white wine vinegar, shallot, black pepper, and 1 tablespoon of tarragon in a saucepan. Simmer until reduced to about 2 tablespoons.
Knead the dough for 8-10 minutes until smooth and cohesive. If dry, add a sprinkle of water; if sticky, add more flour. Shape into a ball, wrap in plastic, and rest for 30 minutes at room temperature.
Prepare floured baking sheets. Divide dough into four, flatten one piece, and roll through a pasta machine starting at the widest setting, gradually reducing the setting with each pass until the desired thickness is reached.