Using an inverted baking sheet in the oven helps give the tart an extra-crisp crust.
Preheat oven to 425 degrees, and place an inverted baking sheet on the middle rack. Brush a 10-by-15-inch rimmed baking sheet with butter.
Brush 1 phyllo sheet with butter, and fit into baking sheet, leaving a 1-inch overhang. (Keep remaining phyllo covered with plastic wrap as you work.) Top with a second phyllo sheet in a slightly different position, and brush with butter. Repeat with remaining phyllo and butter, creating a crust with a 1-inch overhang on all sides. Fold edges under to double the thickness. (Don't worry if some pieces crack along the edge.)
Whisk together eggs, cream, milk, cheese, basil, 1 1/2 teaspoons salt, and 1/4 teaspoon pepper.
Crumple 6 layers of foil into a rectangle the size of the tart, and fit into crust. Bake crust, set on inverted sheet, until edges are golden, 7 to 8 minutes. Remove foil. Bake until surface is golden, 4 to 5 minutes. Transfer crust to a wire rack, leaving inverted sheet in oven.
Immediately pour custard carefully into crust (do not overfill). Arrange zucchini over top, slightly overlapping. Bake on inverted sheet until custard is set in center, 20 to 22 minutes. Immediately slide tart onto wire rack, lifting from 1 end using a large, wide spatula. Let cool for 10 minutes before serving.