How could I easily quantify the amount of EGCG in a sample of green tea?
To analyze specifically this component, you will have to use HPLC. 
In case you can use a more generic approach, you can analyze the total phenolic content with the Folin-Ciocalteu method. 

The Folin-Ciocalteu method is based on a reaction with polyphenols that gives a blue color. 
This color can then be measured in a UV-Vis spectrophotometer and the concentration can be established. 

This method has been used to quantify polyphenols in many plants, as for example here.