Step 1In a large mixing bowl, combine the shallots, vinegar, salt and tarragon leaves. Set aside to steep while you prepare the rest of the ingredients.
Step 2In a small, dry skillet, toast the walnuts over medium heat until they begin to brown and turn fragrant, 3 to 5 minutes. Keep stirring them to keep them from scorching. Set aside to cool.
Step 3Cut each head of endive in quarters lengthwise. Trim the end and the solid core, then cut each quarter into thin, lengthwise slivers. Set aside. Trim the stems from the mushrooms and cut the caps into medium-thin slices.
Step 4When ready to serve, whisk the olive oil and the walnut oil into the vinegar mixture until smooth.
Step 5Place the endive slivers in the dressing and toss to coat well. Lift the endive out of the dressing with tongs and drain excess dressing back into the mixing bowl before arranging the endive in a low mound on a platter.
Step 6Place the mushrooms in the dressing and toss to coat. Using a slotted spoon, drain off excess dressing (taste the mushrooms before draining; they may need a little more salt). Mound the mushrooms on top of the endive.
Step 7Whisk the walnuts in the remaining dressing, drain if necessary, and then scatter them over the mushrooms. Serve immediately.