To make the pasta, combine the flour with the mushroom and umami seasonings and put into a bowl. Make a well in the centre, and pour the beaten eggs. Mix together until it forms a dough, then knead vigorously on a lightly floured surface for a couple of minutes until smooth. Wrap in plastic film and chill for 30 minutes.
For the filling, fry the shiitake mushrooms in the oil in a large frying pan over low heat until the mushrooms release their moisture. Add the soft cheese and combine. Put aside until cool enough to handle.
With a pasta machine roll the pasta mixture to the number 3 setting. Use a circular (8cm) biscuit cutter to cut disks. Put a small amount of filling to 1 side of the disks ensuring enough room is left to seal the ravioli. To seal, wet your finger and run around the outside of the pasta disc then fold over and press the sides together creating a half moon shape.
Bring a pan of water to the boil, carefully place the pasta in the pan. Cook for 4 minutes, until pasta is soft. Carefully remove from the pan.
Combine the oil and pesto and drizzle over the ravioli. Serve immediately.