Combine the arugula and tablespoon of lemon juice in a blender or food processor and process until it forms a fine green paste. Place all the dry ingredients into the bowl of a food processor and pulse to combine. Mix ¼ cup of the arugula paste with the eggs and olive oil. Add the wet ingredients to the dry ingredients and pulse until the mixture begins to form a ball. Add a little water if dough is dry. Remove dough from the bowl, knead until smooth, form into a ball and let rest, covered for at least 1 hour. This dough can be rolled out by hand or to setting 6 on a pasta machine, and then cut into any shape pasta. Bring a large pot of heavily salted water to a boil.
Heat the garlic oil in a large sauté pan set over medium heat. Add minced shallots, and thinly sliced mushrooms and sauté until just barely golden. Add the garlic and sauté for an additional minute. Pour in the white wine and reduce by half. Add the heavy cream and stir to combine. Simmer the sauce for another minute before adding the cheese. Stir until the cheese is completely combined, then taste and season with salt and pepper to taste. Boil the fresh pasta for 2 minutes, drain well, and then add to the sauce in the sauté pan. Simmer for a minute and then serve with additional cheese over the top, if desired.