Why don't my cookies flatten?
From looking at that recipe, those cookies look like they are designed to not spread out.  Why do I say that?  There is a lot of structure in the beginning - the egg whites and butter are beaten together with the sugar and honey for a strong structure made by a hybrid of the creaming method (beating butter with sugar) and the foaming method (beating eggs with sugar) used in cake baking.  There are no egg yolks, and the fat content isn't that high.  Fat helps cookies spread.  So I'd say your cookies are quite possibly doing what they are intended to do.

What you want to do is learn how the various components of your cookie affect its texture - the amount of fat (butter and eggs), the amount of sugar, the method you use to create it, and the temperature all affect your cookies.  You can learn a lot about cookie texture through the questions and answers on this site.


This answer links to a Good Eats episode that talks about three different kinds of chocolate chip cookies, including a soft and thick one.
The question that sparked the above answer has two answers referencing a Cooks Illustrated recipe, their Thick and Chewy cookie.  To get a thick and chewy cookie they melt the butter, add an extra egg yolk, and up the sugar.  A softer, cakier cookie is made by using the creaming method (creaming about 70 degree F butter and sugar).
This answer points you to Michael Ruhlmann's book Ratio, which in the ratio for cookies discusses the varying effects of sugar and fat on your cookies
This question talks about what makes cookies hold their shape versus spread.  One answer suggests the book Bakewise for a comprehensive discussion of gluten content, fat, etc. and how they work in making baked goods.
This answer about how to make cookies without creaming them describes cookies made with melted butter as "pancake cookies" that spread everywhere


As a note, remember that there is also a difference between the temperature your butter is when you make the cookies (especially if you are trying the creaming method versus using melted butter) and the temperature it is at when you cook them.