Warm a large skillet over medium-low heat. Add the butter and melt it, then add the sliced onions and a generous pinch of salt and pepper. Cook for 20 to 25 minutes, stirring occasionally, until the onions are very soft and deeply browned. Set aside.

Position an oven rack in the center and preheat the oven to 450°F.

Crumble both store-bought pie crusts into a large bowl and knead them together until uniform. Roll the dough into a thin large circle or rectangle about 1/8 inch thick, then press it into a shallow quarter sheet pan. Alternatively, divide the dough in half and line two standard pie pans. Sprinkle the fontina, Parmesan, and Gruyère evenly over the dough. Scatter the caramelized onions over the cheese, then top with the cherry tomatoes in a single layer.

Whisk the egg and milk together in a small bowl. Brush the egg wash over the exposed crust edges. Bake for 20 to 22 minutes, watching carefully so the crust doesn't burn; the tomatoes should blister and burst slightly, and the crust should be deep golden brown. If the edges brown too quickly, lower the oven to 425°F.

Remove the tart from the oven and let it rest for 5 minutes. Sprinkle the basil chiffonade over the top. Cut into squares and serve warm or at room temperature.