What is the Proper way to dispose of a broken Monitor?

Look for a e-waste recycling center in your local area.  I think most city's recycling centers should be able to give your the best answer.

I'm pretty sure Best Buys have trucks located in their parking lot to take e-waste now, check their website.

the company I used to work for in florida...thats all we used to do was accept e waste and re sell it or sell the parts and recycle all the other components

smash it in the street and let the homeless collect parts for crack cocaine purposes.