Remove any debris from mushrooms then slice them. Peel the shallots, remove stem ends, and slice thinly.
Add 1 Tb. olive oil to a 12" skillet, over medium heat. Lay mushrooms in pan, ensuring no overlap (if your pan can't accommodate all mushrooms without overlap, make 2 batches - we want to roast, not steam, for maximum flavor). Flip pieces, then remove from pan when brown (make another batch, if needed).
In same skillet, over medium heat, add 1 Tb. olive oil and 1 Tb. unsalted butter. Add shallots and cook until soft/translucent, ~2-3 minutes. Add a little water to the pan to help with softening. When done, add mushrooms and 1 Tb. butter, pinch of salt and freshly ground black pepper. Transfer mixture to a bowl then wipe the pan clean with a paper towel. Cook the eggs according to your preference (scrambled or fried).
Spoon grits, eggs, and mushroom/scallion mixture into 4 serving bowls. Top with prosciutto, herbs and fresh black pepper.