Whenever I put breadcrumbs on chicken breast, they turn out soggy and moist
The classic method of breading ("Wiener Schnitzel") is a three-step process:


flour
Only a thin layer: you can add salt or spices to the flour instead of or in addition to seasoning the meat.
egg
Lightly beaten (not foamy); let excess drip off well. 
breadcrumbs
Either fresh or dried, the egg helps the crumbs to adhere. Lightly pressing the crumbs to the meat is optional.


But there is a fourth component that will ensure your success: fat
Yes, we are all trying to "eat healthy" and monitor our fat intake, but unfortunately fat plays an crucial role in good breaded food. Whether it's deep- or shallow-fried, the hot fat ensures a good heat transmission and quick setting of the breaded crust, thus locking moisture inside.
Your GF grill is an excellent kitchen tool, but defeats the purpose: Your crust can't bake quick enough in the dry heat and touching the hot surface only partially. Combined with the heat from the grill and a significantly longer cooking time, the moisture that steams out of the meat will literally be drawn into the crust - leaving you with sub par results.

Save your GF for other uses and use a pan with some oil to fry your chicken. 



Side note:
Some recipes for fried chicken recommend baking the floured chicken pieces in an oven to save fat. Note that a fried chicken is typically a two-step coating with an outer layer of flour, so slightly other rules apply.

Nevertheless, this method might be an option for you and worth a try, but don't expect the golden crust you get from traditional frying. I have used an oven with breaded meat, but typically with partially fried food, so that the breading already contained fat / oil from the first round of frying.