Whisk together 1 teaspoon olive oil, water, and whole eggs in a bowl until evenly blended. Combine flour and salt in a separate large bowl, making a well in the center. Add the egg mixture into the well and stir until combined. Knead on a floured surface until smooth, about 5-10 minutes. Wrap dough tightly with plastic wrap and let rest.

Heat olive oil in a skillet over medium heat. Add minced garlic and chopped onion. Cook until softened, about 2 minutes.

Add chopped mushrooms to the skillet and cook until softened and liquid has evaporated, about 10 minutes. Remove from heat and cool.

In a bowl, beat cream cheese until smooth. Stir in mushroom mixture, Parmesan, mozzarella, thawed spinach, chives, parsley, cayenne, and seasoning. Mix thoroughly.

Roll out dough to 1/16 inch thickness. Cut large circles using cookie cutter. Brush each circle with egg white. Place filling in the center of one circle, cover with another circle, and seal edges. Repeat.

Bring salted water to a rolling boil in a large pot. Cook ravioli until they float to the top, about 3-4 minutes. Drain and set aside.

Melt butter in a skillet over high heat until browned, stirring frequently for 5–7 minutes. Add chives and remove from heat.