Make the flaky pie dough: Pulse the flour, salt, and mayonnaise in a food processor until crumbly. Gradually add the cold water through the food chute and pulse just until the dough comes together. Shape into a disk and let rest at room temperature for 30 minutes. (Or, refrigerate until ready to use and let stand at room temperature for 1 hour or until pliable.)
Preheat the oven to 400°F. Roll out the pie dough to a ¼-inch thickness. Fit into a 9-inch pie plate or tin; fold the edges until and crimp. Fill the pie shell with parchment paper and pie weights or dried beans. Bake at 400°F for 12 minutes, or until the crust is baked through. Remove the pie crust from the oven and let cool. Reduce the oven temperature to 350°F.
Heat the oil in a large nonstick skillet over medium-high heat. Add the garlic and spinach and cook for 2 to 3 minutes, until the spinach wilts.
Spread the spinach mixture and tomatoes evenly in the pre-baked pie crust. Whisk together the eggs, mayonnaise, milk, Asiago cheese, basil, salt, and pepper in a medium or large mixing bowl. Pour over the spinach mixture.
Dot the filling with the goat cheese. Bake the quiche at 350°F for 40 minutes, or until the quiche is set. Let cool on a wire rack before cutting into wedges. Garnish with basil leaves, if desired.