Melt butter, cubed, in a saucepan over medium-high until just browned, 2 to 3 minutes. Reduce heat to medium. Add finely chopped shallot; cook, stirring constantly, until butter is golden brown, about 1 minute. Whisk in uncooked coarse-ground grits; cook, stirring constantly, until butter is absorbed, about 1 minute. Whisk in water, and bring to a boil over medium-high, stirring often. Reduce heat to low; simmer, stirring often, until thickened, about 45 minutes. Remove from heat; stir in Parmesan cheese, grated, heavy cream, and kosher salt.