How can I make use of an underripe mango
If you have a WHOLE, UNCUT green mango, you can ripen it on the counter.  To speed this process, the mango may be place in a paper or partially-sealed plastic bag; this will help retain the ethylene gas fruits emit, which speeds ripening.  

Once cut, you'll have to put your mango to use. Fortunately, green mangoes are a prized part of Southeast-Asian cooking, where their tartness is used to add tang to savory dishes!  In this use, green mangoes are often combined with salt, chili peppers, sometimes lime juice, and sometimes soy or fish sauces.  Coconut milk is another common flavor combination, for example coconut rice topped with green mango preserves. 

In particular, Thai cuisine makes an extensive use of green mangoes.  For an example, look at this green mango salad recipe.  In India, they may be sliced and topped with salt and lime juice, or pickled, or incorporated into a delicious green mango chutney.  Indians also grind dried green mango into amchur, a sour powder used in curries and other dishes.  

Central American cuisine also includes green mangoes, and they serve it sliced with salt, vinegar, pepper, and hot sauce.  Topping with toasted pumpkin seeds is also common. 

:Edit:
The one common thread here is that green mangoes are generally used in savory, not sweet dishes.  This is because the flavor of under-ripe mango is quite different from that of ripe ones. This different flavor works best with salty and spicy combinations, with (brown) sugar as a secondary, background note, not as a dominant flavor.  I would particularly discourage combining green mango with milk, unless you want the acid to curdle it into curds and whey.