Heat a large skillet over medium-low heat. Melt the butter, add the sliced onions, and season with a pinch of salt and pepper. Cook for 20 to 25 minutes, stirring occasionally, until the onions are soft and deeply golden brown. Remove from the heat and set aside.

Preheat the oven to 450°F and adjust a rack to the center position.

On a lightly floured surface, press the two pie crusts together into a ball and knead a few times to combine. Roll the dough into one large, thin crust and lay it onto a shallow quarter sheet pan. Alternatively, cut the dough in half and fit it into two standard 9-inch pie pans. Sprinkle the Fontina, Parmesan, and Gruyère evenly over the crust, then spread the caramelized onions over the cheese, and scatter the cherry tomatoes over the top.

In a small bowl, whisk the egg and milk together. Brush this mixture over the exposed edges of the crust. Bake for 20 to 22 minutes, watching closely, until the tomatoes start to blister and burst, the cheese is melted, and the crust is deep golden brown. If the crust is browning too quickly, lower the oven temperature to 425°F.

Remove the tart from the oven and let it rest for 5 minutes. Sprinkle the sliced basil over the top, cut into squares, and serve.