Whisk together 1 teaspoon olive oil, water, and whole eggs in a bowl until evenly blended; set aside.
Combine flour and salt in a separate large bowl, and make a well in the center.
Pour the egg mixture into the well and stir just until combined. Turn dough out onto a lightly floured surface and knead until smooth, 5 to 10 minutes, adding more flour or water as needed. Wrap dough tightly with plastic wrap, and set aside to rest.
Heat 1 teaspoon olive oil in a skillet over medium heat. Stir in the garlic and onion; cook and stir until the onion begins to soften, about 2 minutes.
Add the mushrooms, and continue cooking and stirring until the vegetables are soft and the liquid has evaporated, about 10 minutes.
Remove from heat, and allow to cool.
Beat cream cheese in a bowl until smooth. Stir in the cooled mushroom mixture, Parmesan cheese, mozzarella cheese, spinach, 1 tablespoon chives, parsley, and cayenne pepper. Season with salt and pepper.
Roll the pasta dough out to about 1/16 inch thick.
Cut 3 to 4-inch circles using a large cookie cutter.
Roll each circle out as thin as possible. Working with one circle at a time, brush the pasta lightly with the egg white. Scoop about 1 heaping tablespoon full of the mushroom filling onto the center of the pasta, then cover with a second piece of pasta, pinching the edges to seal.
Cut the sealed ravioli with the cookie cutter once more to create a uniform shape.
Place the finished ravioli on a floured baking sheet, and repeat the process with the remaining pasta and filling.
Fill a large pot with lightly salted water and bring to a rolling boil over high heat. Once the water is boiling, stir in the ravioli and return to a boil. Cook until the pasta floats to the top, 3 to 4 minutes; drain.
To make sauce: Melt butter in a skillet over high heat, cooking and stirring until browned, 5 to 7 minutes. Stir in 1 1/2 teaspoons chives.
