Place a large skillet over medium-low heat. Melt the butter, then add the onions, 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 degrees Fahrenheit.

Press both pie crusts together into one ball, knead briefly to combine, then roll out into one large, thin sheet. Lay the dough into a shallow quarter sheet pan, or cut in half and press into two standard pie pans. Sprinkle the fontina, Parmesan, and Gruyere evenly over the dough, then scatter the caramelized onions on top, and finish with the cherry tomatoes.

Whisk the egg and milk together in a small bowl, then brush the mixture over the exposed crust edges. Bake for 20 to 22 minutes, checking often so the crust does not burn. The tomatoes should start to burst and develop some dark, roasted spots, while the crust turns deep golden brown. If the edges brown too quickly, lower the oven to 425 degrees Fahrenheit.

Let the tart rest for 5 minutes out of the oven. Sprinkle the basil ribbons generously over the top, cut into squares, and serve warm or at room temperature.