Trim picanha steak by removing silver skin from non-fat cap side. Trim off any excess fat pockets from fat layer if it is not firm, but leave at least 1/4 inch of fat on.
With fat side of steak down, cut into 1 inch to 1 1/4 inch steaks buy cutting perpendicular to the grains on the steak. Should yield four to five steaks.
Mix dry rub ingredients together in a small bowl and apply to the steaks.
Place steaks on direct side of heat for 6 - 8 minutes and then flip. Continue cooking for another 4 - 6 minutes. Move to indirect side to finish until desired temperature. We like to pull when internal temperature of the steaks is rare or reads between 120 - 125 degrees F.
Let rest for 15 minutes. This is a good time to make the mushroom cream sauce.
Mushroom Sherry Cream Sauce
Prepare all ingredients prior to grilling the steaks. Remove stems from mushrooms. Using damp towel or paper towel, gently brush and remove any excess dirt from mushrooms. Slice mushrooms thin.
Over medium high heat in a large skillet, add butter and as it melts add the mushrooms. Stir mushrooms occasionally to brown. Remove mushrooms when browned and set aside, leaving some of the juices/liquid. About 8 - 10 minutes.
In same skillet over medium heat add the garlic. Sauté garlic for two minutes using residual liquid from mushrooms. Add sherry and stir, cooking for 1 - 2 minutes.
Add cream, stock, cheese, salt, pepper, and thyme and bring to a simmer. Add mushrooms and reduce heat slightly. Make slurry with the extra stock and flour in a small jar if the sauce isn't to your desired thickness. Shake and pour into the simmering liquid. Let simmer for additional 6 - 8 minutes, it will thicken while simmering. Remove from heat.
Serve over steaks.
*If you only have a small pan - Saute the mushrooms in two batches, be sure to also portion the butter.