Torte Milanese without puff pastry dough
If you use something other than puff pastry it's not a Torte Milanese anymore, and other kinds of pastry will not hold together in the same way puff pastry does. Puff pastry's layers provide a certain amount of strength and will help keep the shape when sliced, other pastries will crumble much more easily. You can use short crust or any other pastry, it won't be the same but maybe that's not so important to you.

You could of course just buy pre-made puff pastry, there's no shame in that. Puff pastry is challenging and time-consuming to make, and in many places you can buy frozen pre-made all butter puff pastry which gives excellent results. I've made puff pastry, but when I need it I usually buy it as I don't have hours free to spend on it. 

One option would be to make "rough puff" pastry, also known as "flaky pastry" rather than full-blown puff pastry. It's less time consuming, a bit more forgiving, and will still give a similar product.