Using a microplane, grate eggs into a medium bowl. (Alternatively, press eggs through a fine-mesh sieve set over a medium bowl.) Add onion, mayonnaise, mustard, chives, dill, nutritional yeast, pepper and salt; gently fold together until just combined and creamy.
Spoon about ½ cup of the egg salad onto each of 2 bread slices; top each with 1 ½ tablespoons Parmesan. Top with the remaining 2 bread slices; press down gently on each sandwich.
Melt butter in a large nonstick skillet over medium-high heat; swirl to coat. Add the sandwiches; cook, flipping once, until the bread is golden brown on both sides, about 4 minutes.
