For the chive oil, place the 1/2 cup of walnut oil, chives, and pinch of salt in a blender and blend until smooth. Let the mixture drip through a fine sieve to remove the solids.
For the shallots, break the slices of shallot up into rings. Coat the bottom of a skillet with avocado oil and heat over medium-high heat. Cook the shallots, stirring frequently, until they are well browned. Let them drain on paper towels and sprinkle lightly with salt.
Preheat the oven to 350 degrees. Butter four, 8 ounce ramekins. Divide the mushrooms evenly between the ramekins.
Whisk together the eggs, egg yolks, cream, salt, and white pepper. Pour the mixture over the mushrooms in the prepared ramekins. Place the ramekins in a baking dish and pour about an inch of hot water around them. Bake for 40-50 minutes, or until the centers no longer wiggle when you shake the ramekin.
To serve, run a sharp knife around the edge of each ramekin. Place the serving dish on top of the ramekin and carefully flip it to invert the flan. Garnish each flan with some of the chive oil and some crispy shallots.