Preparation

Make the crust

In a large bowl, whisk together the flours, baking powder, and salt. Transfer to a food processor, and add the butter. Process until coarse crumbles form, about 35 seconds. Add the sour cream, and process until a smooth dough forms, about 20 seconds. Transfer the dough to a lightly floured surface, and shape into an 8×10-inch rectangle. Fold into thirds, sprinkle with flour, and roll out to an 8×10-inch rectangle. Fold into thirds, rotate 90 degrees, and roll out to an 8×10-inch rectangle again. Wrap in plastic wrap, and let chill at least 30 minutes or overnight if needed.

Smoke the mushrooms

Prepare a charcoal chimney, heating charcoal until ashed over and glowing. Meanwhile, soak wood chips in water at least 20 minutes. Be sure the bottom vents are just open. Gently pour the heated charcoal in an even layer. Drain the wood chips and
carefully sprinkle over the charcoal. They will begin to smoke immediately. Add the grilling grate. Place the mushrooms in a disposable foil pan on the grill grate, and cover the grill with the top vents slightly open. Smoke for 10 to 15 minutes. The mushrooms should still be soft and some liquid will have accumulated. Remove from the heat and let cool.

Assemble and bake the tart

Slice the smoked mushrooms into thirds, keeping the stem intact, and set aside. Slice the tomatoes into 1/4-inch rounds, removing any pulp. Place on paper towels, and sprinkle lightly with salt. Let stand 10 minutes to help remove any excess liquid.

Preheat the oven to 425°F. Remove the pie crust from the refrigerator. Between two sheets of parchment paper, roll the pie crust to a 17×12-inch rectangle (crust will be thin). Remove the top sheet of parchment from the dough, invert the crust into a 15×10- inch baking pan, and gently remove the bottom sheet of parchment. Ease the crust into the pan and gently press it into edges of pan, allowing it to hang over the edges as needed. Trim evenly with the edge of the pan. Using a fork, prick the dough all over. Cover with a sheet of parchment, and place a 13×9-inch pan, pie weights, or beans over the crust. Bake until nearly done, about 15 minutes. Remove the parchment, baking pan, pie weights, or beans. Let stand 10 minutes. Decrease the oven temperature to 400°F.

Meanwhile, in a large bowl, whisk together the eggs, mayonnaise, sour cream, pepper, and mustard until combined. Fold in the cheese and basil. Using an offset spatula, spread the filling over the crust. Add the tomato and mushroom slices, pressing gently into filling. Bake until filling is just set and light golden brown, about 20 minutes. Remove from the oven. Let stand 20 minutes before slicing. Garnish with basil leaves.