In a small bowl, add the egg and slightly whisk. Add water and oil. Whisk until incorporated and set aside. In another bowl combine the semolina flour and all-purpose flour.
Add the beaten egg mixture to the flour and stir to combine. Mix to make a stiff dough. Turn out onto a lightly flour dusted flat surface and knead 10 minutes until smooth. Test by pressing your knuckle into the dough, if it starts to bounce back then it’s ready.
Flatten slightly into a disk shape. Wrap dough in plastic wrap and set aside on the counter (do not chill) for at least 30 minutes.
Dust your work surface with semolina flour. Cut the pasta dough into 4 pieces. Working with one piece at a time, flatten the dough into a 6-8-inch oval patty. Keep the remaining dough covered in wrap otherwise they will dry out.
Using a pasta roller, set to #1 which is the lowest/widest setting. Send the dough through the roller twice. Roll the dough as thin as you can, about 1/16 inch.
Continue to roll the dough through settings #2-4, (2x) on each setting. Do not pull on the sheet as it comes through the roller; just guide it along gently. If at any point the dough becomes sticky, dust with semolina flour.
Your pasta sheet is ready when it’s thin and somewhat translucent (you should be able to see your hand underneath).
To finish, dust the pasta sheet generously with semolina, fold it in half, and place it onto your floured sheet pan. Cover with a kitchen towel. Repeat the rolling process for the remaining dough.
Take a pasta sheet and cut it in half. Cut the dough into strips about 12-inches long by 1-inch thick. Transfer the cut pasta to a floured baking sheet, dust/rub the pasta with a bit of flour, and swirl into little nests.
To Cook the Pappardelle
Bring a large pot of generously salted water to a boil. Cook the pasta until al dente. Drain and set aside.
For the Sauce
Add the olive oil to a large skillet and heat over medium-high heat. Add the garlic and yellow onion and sauté together for about 3 minutes.
Add the artichoke hearts, lemon juice, white wine vinegar, salt, pepper, and vegetable broth. Bring to a simmer for about 10 minutes, or until the liquid has reduced to half and coats the back of the spoon.
Stir in half the basil. Add the butter and let it melt into the sauce. Stir.
Toss the pasta in the pan with the sauce and remaining basil.