How do you convince a professor to write you a recommendation letter, even though it has been more than 10 years since you counted as one of his students?

I have reached a virtual dead-end in my professional career, and without further education, I will not be considered for any meaningful positions. Hence, I have decided to pursue a PhD. As you all well know, at least two well written recommendation letters from academia are a must for any successful application.

The problem is that I am relatively old now (37), and have not had any contact with my lecturers in graduate school for almost 10 years. Some of my professors still teach at my old University, though I doubt any of them, except for my thesis advisor really remembers me. What I do got going for me is that I can show for a strong Master's degree, having graduated summa cum laude, 10 years ago.

How would you even approach a potential recommendation letter writer? Who would you choose and why?