ETYMOLOGY
from grout (sediment, dregs, lees), taken as the type of something big and coarse;
for definition 2: there is confusion with great; perhaps the sound recalled the Dutch groot
EXAMPLE
“…professed her self a nonne in the yeare of our lord a. M. and. lxxv. to serue the deuyll in the monkes hypocresy, & in ye burnynge heates of Sodome. So daynty mowthed wer these greasy grouteheades, and so crafty in their generacyon, that they could fynde out kynges doughters to serue their lustes, and yet apere chast ghostly fathers to the world. Thurstinus a monke of Cane in Normandy…”
From: The First Two Partes of the Actes or vnchast examples of the Englysh Votaryes gathered out of their owne legenades and chronycles
By John Bale, 1551
Lexicophilia 