Place the mushrooms in a medium mixing bowl and toss gently with the salt. Place a plate on top of the mushroom and weigh it down with a heavy object. Let the mushrooms sit for an hour. Drain, spread them on a clean towel and pat them dry.
Bring the vinegar to a boil in a medium saucepan over medium-high heat. Add the mushrooms, return to a boil, and boil until they have softened slightly but still have a nice meaty texture, 3-5 minutes. Drain and spread the mushrooms out on a clean towel and pat them dry. Let them dry until they are no longer damp, about 2 hours.
Combine the oil, mustard seeds, tarragon and lemon zest in a small saucepan over medium heat. Heat, stirring from time to time, to just below a simmer (look for small bubbles to appear). Remove from the heat and let cool slightly.
Place the mushrooms in a wide mouth pint-sized canning jar. Pour the infused oil over the mushrooms, adding more oil to submerge completely if necessary. Cover and refrigerate. To serve, bring to room temperature and remove only what you plan to use. Top the jar with oil as necessary to keep the remaining mushrooms submerged. Eat within 3 months.