Rinse and drain mushrooms. Trim off and discard tough stem ends and any bruised areas; finely chop mushrooms.
In a bowl, beat eggs to blend. Add mushrooms, rice, cheese, bread crumbs, green onions, hot sauce, thyme, salt, and pepper; mix until well blended.
Heat 1/2 tablespoon olive oil in a 10- to 12-inch frying pan over medium-high heat. Spoon rice mixture in 1-tablespoon portions, slightly apart, into pan; with back of spoon, flatten into 3-inch rounds. Cook pancakes, turning once, until browned on both sides, 4 to 6 minutes total. With a spatula, transfer to a 12- by 15-inch baking sheet lined with paper towels and keep warm in a 200° oven. Repeat to cook remaining pancakes, adding more oil to pan as needed.
Arrange pancakes in a single layer on a platter. Dollop sour cream equally on pancakes. Top each with 1/2 teaspoon salmon roe if desired.
Nutritional analysis per pancake.