Make the crust
Combine both flours and the salt in a food processor; pulse to combine. Add the butter and pulse until it breaks down to the size of small peas. Sprinkle 4 Tbs. of the ice water over the mixture and pulse again until the pastry just holds together (if it’s too dry to hold together, pulse in tiny amounts of the remaining water until it holds). Transfer the dough to a large sheet of plastic wrap, and using the plastic as an aid, shape it into a thick disk. Wrap the dough in the plastic and refrigerate for about 30 minutes.
Meanwhile, position a rack in the center of the oven and heat the oven to 375°F.
Unwrap and roll the dough on a lightly floured surface into a 14-inch circle. (If the pastry cracks, just press it back together.) Wrap the pastry around the rolling pin and unroll it over an 11-inch fluted tart pan with a removable bottom. Without stretching the dough, very gently work it into the pan, pressing the pastry against the sides. Roll the pin over the pan to trim the excess dough. Use the scraps as needed to make the edge even and about 1/4 inch thick (at the narrowest points). Press gently all around the edge so the dough comes up slightly above the rim of the pan.
Prick the bottom of the crust all over with a fork, line with parchment, and fill with dried beans or pie weights. Put the tart pan on a baking sheet and bake until the edge looks dry, about 10 minutes. Carefully remove the beans and parchment and bake until the bottom is just set and looks dry, 5 to 7 minutes more. Cool on the baking sheet on a wire rack.
Make the filling
Scrub the sweet potatoes, poke them once or twice with a fork, and put them on a small foil-lined baking sheet. Roast until tender when pierced, 50 to 60 minutes. Cool, peel, and cut into 3/4-inch dice.
While the sweet potatoes are roasting, heat 2 Tbs. of the olive oil in a 12-inch nonstick skillet over medium-high heat. Add the onions and a generous pinch of salt; cook, stirring, until the onions wilt and develop dark-brown charred spots, about 10 minutes. Reduce the heat to medium low, cover the pan, and stir frequently, until softened and caramelized, 8 to 9 minutes more. Add 1 Tbs. of balsamic vinegar and stir until it evaporates and glazes the onions.
Transfer the onions to a small bowl and wipe out the pan. Heat the remaining 1 Tbs. olive oil in the pan over medium-low heat. Add the garlic and cook until fragrant and just beginning to color, 1 to 2 minutes. Increase the heat to medium high, add the chopped chard and a pinch of salt, and toss over medium-high heat until the chard is completely wilted, about 4 minutes. Sprinkle the remaining 1 Tbs. balsamic vinegar over the chard and toss it until the vinegar cooks away, about 1 minute. Transfer to a large bowl.
Add the diced sweet potatoes, about three-quarters of the cheese, and a pinch of crushed red pepper to the chard; toss gently. In a small bowl, whisk the eggs with a pinch of salt and pepper and add the eggs to the chard mixture.
Bake the tart
Spread the filling evenly in the tart shell and scatter the remaining cheese on top. Bake the tart on the baking sheet until the cheese is nicely browned, 25 to 30 minutes.
Spoon the balsamic onions over the top of the tart, allowing bits of cheese to peek through here and there. Let the tart cool slightly, about 10 minutes; then remove the rim of the pan. Slice and serve the tart warm or at room temperature.