For the crackers: Preheat the oven to 400 degrees F. Line a baking sheet with parchment paper.
Combine the flour, cornmeal, Parmesan, tarragon, cayenne and 1 teaspoon salt in a food processor and pulse to combine. Add the butter and pulse until incorporated. Add the heavy cream and process until the dough comes together. If the dough is too dry, add a little more cream, 1 teaspoon at a time, until the dough comes together.
Lightly flour a sheet of parchment paper and place the dough on top. Shape the dough into a rectangle with your hands, then lightly flour the dough and roll it out with a rolling pin until it is a little less than 1/4 inch thick (about a 15-by-12-inch rectangle). Poke all over with a fork. Cut the dough into squares or rectangles and carefully transfer them to the prepared baking sheet. Bake until light golden brown, 10 to 12 minutes. Cool completely on a rack.
For the mushrooms: Increase the oven temperature to 450 degrees F. Line a baking sheet with foil.
Toss the mushrooms with the oil, 3/4 teaspoon salt and a few grinds of pepper on the prepared baking sheet. Roast, stirring once or twice, until tender and well browned, 20 to 25 minutes.
In a small bowl, combine the lemon juice, tarragon and a pinch of salt and pepper. Transfer the mushrooms to a bowl and top with the tarragon-lemon juice mixture.
Serve the crackers with the mushrooms on the side for topping.