Much of the binding in the filling comes from milk and cheese, so I decided to use silken tofu and miso as a replacement. The former replaces the creaminess and the latter replaces the cheesiness. Yes, you read that right - miso can imitate a cheesy flavor. It's a trick I learned from the meatball recipe on the blog Supper for Two.
Another big change I made to this recipe is the cooking time, as the tofu takes much longer to set than cheese. While cooking, I was getting nervous about leaving in the tart for nearly double the original cooking time, but it turned out perfectly.
In the end, this tart came out so delicious, meaty from the mushrooms and savory from the miso. We had absolutely no leftovers, which is always the best sign of all.