How to substitute butter with combination of butter and shortening in cookie/biscuit making?
The texture of a cookie is based on much more than the fat used, shortening or butter.  In fact, within some basic limits, they are fairly interchangeable in most cookie recipes, flavor not withstanding.

Switching to part or all vegetable shortening will not yield a flaky texture.

The method by which the ingredients are combined, and how the cookies are treated, mixed, or rolled is a dominant factor in the final texture.  

As Sourd'oh points out, your variation is more likely to be a result of over-cooking or under-cooking.   The individual size of each cookie can make a considerable difference, especially with very small cookies.

If you are getting inconsistency within the same tray, you may not have uniformly sized cookies.  A cookie scoop or disher can help with that, as can practice.

To get a truly flaky texture, you would need to use a recipe and method specifically designed to create flakiness.  These cookies are often built with a variation on the laminated methods used for biscuits, where butter is cut into the dough, and then moistened.   When they are rolled out, the pieces of butter flatten into layers, separating the flour layers, and providing the flakiness.