Preheat the oven to 190c. Grind the bread to breadcrumbs in a food processor. Butter and breadcrumb the inside of the soufflé dish, knocking out any excess. My dish was 20cm wide and 8cm high - with this size of dish, the soufflé will rise about 3cm above the edge. If you want a higher, more spectacular rise, use a dish with a smaller volume.
In a medium saucepan, whisk together the milk, cream and rye flour (sprinkle the flour in gradually to avoid lumps). Add the mushrooms, garlic, paprika and black pepper and heat over medium heat until very well thickened. Add the fresh basil and salt. Taste a little of the mixture and add more salt if you think it needs it - it should be fairly salty to allow for the addition of egg whites - and stir to combine. Transfer to a large mixing bowl and set aside to cool for about 10 minutes.
Separate the eggs, adding the yolks to the mushroom mixture and the whites to a very clean bowl for whisking (the bowl you choose will depend on your whisking method and whether you are using a machine or not.) Beat the yolks into the mushroom mixture until thoroughly incorporated. Whisk the whites to stiff peaks (I used a stand mixer). Gently fold the whites into the mushroom mixture - there should still be lots of air in the batter, but there should be no large blobs of egg white.
Pour into the prepared dish - if necessary, the uncooked soufflé will hold for about an hour after it has been added to the dish. Bake at 190c for 35 minutes, until well risen and firm, but still moist and creamy in the centre. Serve immediately.