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.
In a small mixing bowl, whisk together all ingredients expect the blue cheese until fully combined. Fold in the blue cheese, making sure to leave some pieces in larger chunks for added texture. Taste and adjust seasoning, adding more salt, sugar or acid to balance your particular cheese.
This has a nice garlic kick at the end which I love with the other flavors. If you don't like the lingering taste of raw garlic, replace it with a little minced shallot.
Heat the olive oil in a 12" skillet over medium heat until shimmering. Add the shallot and cook, stirring frequently, until softened, about 2-3 minutes. Add half of the cherry tomatoes, and cook, stirring frequently, until they have softened and have just started to break open, another few minutes. Remove from heat and carefully transfer to a blender. Add the remaining ingredients, including the other half of the cherry tomatoes and starting with just 1 teaspoon of white wine vinegar. Blend on high until a smooth sauce forms, about 1 minute. Taste and add another teaspoon of white wine vinegar if the sauce needs a little extra zing. Transfer sauce to a freezer safe container, leaving at least 1" headspace if planning to freeze, and cool to room temperature. Store in the refrigerator for 1 week or the freezer for up to 1 year.