How do doctors measure how thin my blood is?

Thinning Agent??? I assume that you are referring to the use of a "blood thinner", such as Coumadin (warfarin). In this case, we do a blood test called the prothrombin time (pt). If you are using Heparin as a blood thinner, we would check your activated partial thromboplastin time (aptt). I hope this is what you were referring to?
Viscosity? Viscosity isn't typically as important for most patients, compared to "content", such as the number of red blood cells, platelets, etc.