How do I make a flaky croissant?
Flakiness in pastry is usually achieved by careful incorporation of butter at the correct temperature. Cold cubes of butter are cut into the flour, cold water (or milk) is added, and then the pastry is rolled out, flattening the butter pieces. These pieces act to separate layers of the flour and liquid mixture.

The butter then melts during cooking, 'setting' these layers in the dough. The outer surface of the dough dries out, which means the layers don't stick together, and so it becomes flaky.

If the butter becomes warm and soft during rolling or cutting in, it blends with the flour and liquid and so doesn't promote layer formation.

In my experience, even refrigerated shortening is usually quite soft, and may well be simply absorbing into the rest of the dough when you roll it in. You might try freezing the shortening until it reaches the same consistency as refrigerated butter and working from there. I would also use ice cold water rather than warm water as the recipe suggests.