Bring a pot of water to a boil and fill a bowl with ice cubes and water. Blanch the basil in the boiling water for 10 seconds, then refresh in the ice water. Drain and chop finely; you should have 1/4 cup. Make a mound of the flour, then make a well in the center. Crack the eggs into the well, pour the milk over and add the basil. Using a fork, stir the egg mixture slowly into the four ingredients to form a wet dough. Add the grated cheese and, working now with your hands, bring the dough together and knead for 8 to 10 minutes to form a smooth dough. Allow to rest for 15 minutes covered with plastic wrap.