In a food processor, pulse the whole-wheat and all-purpose flours with 1/2 teaspoon of salt. In a bowl, whisk the egg yolks, water and the 1 teaspoon of oil. With the processor on, add the egg yolk mixture and process until a ball forms. Transfer the dough to a work surface and knead until smooth. Cover with plastic wrap and let stand at room temperature for 20 minutes. Wipe out the processor.
In a small skillet, toss the pine nuts over moderate heat until toasted, 1 minute; let cool. In the food processor, pulse the arugula, pine nuts, cheese, garlic and the remaining 1/3 cup of oil to a paste. Season the pesto with salt and add to a large skillet.
Cut the dough into 3 pieces; work with 1 piece at a time and keep the rest covered. Flatten the dough slightly and run it through progressively narrower settings in a pasta machine until you reach the thinnest setting. Cut the pasta into 10-inch lengths. Using the pasta machine, cut each length into fettuccine. Transfer to a baking sheet, leaving space between the pasta so it doesn’t stick.
In a pot of boiling, salted water, cook the pasta, stirring, until al dente, about 30 seconds. Drain the pasta, reserving 1/3 cup of the cooking water. Stir the reserved water into the pesto. Add the pasta and tomatoes to the pesto, toss over moderate heat and serve.