Step 1 Peel and dice the parsnips. Boil in salted water until tender, about 4 minutes. Drain.
Step 2 Pour the half-and-half and heavy cream over the parsnips in the same saucepan. Bring to a simmer, then remove from heat and transfer to a blender. Puree until smooth. Pour the mixture into a large bowl. Add salt to taste.
Step 3Whisk one egg yolk at a time into the parsnip-cream mixture.
Step 4Heat the oven to 300 degrees. Use 1 tablespoon butter to grease 8 (4-ounce) ramekins. Divide the parsnip mixture evenly among the ramekins. Place them in a 13-by-11-inch baking pan. Pour hot water into the pan until it comes halfway up the sides of the ramekins. Cover the entire baking pan with foil and bake until the flans are set, about 40 to 45 minutes.
Step 5Remove the pan from the oven. Uncover the entire pan, but leave the ramekins in the water for about 15 minutes to set completely. Invert each ramekin onto an individual plate, tapping lightly to unmold. If the flans are stubborn, run a knife around the edges to remove.
Step 6Saute the mushrooms and shallots in the remaining butter until soft and golden brown, about 15 minutes. Season with salt and freshly ground black pepper.
Step 7To serve, spoon the mushrooms and shallots over the flans and garnish with fresh chervil.