A-overhang
Adenylation of blunt-ended DNA fragments enables you to create a binding site for a specific adapter to any DNA strand. Adenylation of a blunt-ended DNA results in a 3' end Adenine (A) overhang, and as you remember, Adenine pairs up specifically with Thymine.
The adapters have thymine at the 3' end that will specifically bind to the 3' adenine of the DNA sample.
A non-proofreading polymerase such as Taq could be used to add an adenine at the 3' of the blunt end DNA.
Figure 1. A-overhang