Wash and dry mushrooms and slice off stems. Finely dice mushrooms.
Whisk eggs and stir in mushrooms, rice, cheese, bread crumbs, onions, hot sauce, thyme, salt, and pepper.
Pour 1/2 tablespoons oil in a frying pan and heat over medium-high heat. Spoon rice mixture into pan in 1 tablespoon portions, slightly apart and flatten with the back of a spoon. Cook pancakes 2 to 3 minutes per side. Move pancakes to a platter lined with paper towels and keep warm in a 200 degree F oven. Cook the rest of the pancakes in the same manner.
Lay pancakes in one layer on a platter and top with sour cream and salmon roe.