Special equipment: pasta rolling machine or pasta rolling attachment for a stand mixer
Place the cake flour, all-purpose flour, egg yolks and salt in a food processor. Pulse to combine. With the machine running, gradually add the oil then 1/3 to 1/2 cup water until the mixture forms a dough (the dough should stick together if pinched between your fingers). If necessary, add additional water, 1 teaspoon at a time if the dough is too dry.
Place the dough on a lightly floured surface. Gather the dough into a ball and knead until the dough is smooth, 5 to 8 minutes. Cover with plastic wrap and refrigerate for 30 minutes.
Cut the dough into quarters and press flat. Run each piece of pasta dough several times through a pasta-rolling machine, adjusting the setting each time, until the pasta is about 1/8 to 1/16-inch thick. Cut the pasta into the desired shape and use as needed.
The pasta dough can be rolled out by hand. Cut the chilled dough into quarters and using a rolling pin, roll out the dough until 1/8- to 1/16-inch thick.