Place the flour, olive oil and eggs in a food processor and blitz until large crumbs begin to form.
Transfer onto a lightly floured work surface and knead for 2 minutes to obtain a smooth and even consistency. Wrap in clingfilm and refrigerate for 20 minutes.
Butternut squash and girolle filling
Dice the butternut squash into 1 cm cubes. In a pan, sweat the butternut squash with half the butter over a low heat. Once soft, about 10 minutes, remove 50 g of the squash and reserve for plating. Remove the remaining butternut squash from the pan and crush with a fork.
Wash and dry the girolles, then pan-fry in the remaining butter until soft, about 5 minutes. Allow to cool, then roughly chop half the girolles and combine with the crushed butternut squash. Season to taste. Reserve the remaining girolles for plating.
Butternut squash and parmesan cream
Roughly chop the squash into small pieces and transfer to a saucepan with the milk. Bring to a simmer and cook until soft, about 10 minutes. Grate the parmesan and add to the mixture, then blend using a stick blender until smooth and frothy. Season to taste and keep warm.
Using a pasta machine, roll out the dough until it is 2 mm thick. Cut out into discs with a diameter of 6 cm. Place a small amount of the butternut squash and girolle filling in the centre of each disc. Brush the edges of each disc with the beaten egg and fold over to make a half-moon shape. Take the folded edges of each half-moon and pinch together into the centre to form a round shape.
In a pan, melt the butter over a low heat. Reheat the reserved girolles and butternut squash. Finely chop half the sage leaves and add to the pan with the capers.
Shallow fry the remaining sage leaves in the olive oil until crisp, about 2 minutes. Lift the sage leaves out of the pan and drain on kitchen towels. Season with salt.
Bring a pan of salted water to a boil and cook the tortellini for 3 minutes. Arrange 5 tortellini on each plate and top with the girolles, butternut squash, sage and capers. Spoon the butternut and parmesan cream over the dish. Finish with the parmesan and fried sage leaves.