Beat eggs with 3/4 cup plus 2 tablespoons mayonnaise in a bowl. Add Gouda cheese and flour; mix well. Pour 1/2 the batter into another bowl.
Heat 1 tablespoon oil in a skillet over medium heat and cook grated carrot while stirring until softened, about 5 minutes. Add the carrot to 1 portion of batter; mix a small bunch of dill into the other portion.
Heat a nonstick or well-seasoned cast iron skillet over medium-low heat. Pour in 2 tablespoons of one of the batters; immediately rotate the pan in a circular motion to spread. Cook until edges are crispy, 1 to 2 minutes. Flip until browned and dry on the other side, 1 to 2 minutes more. Repeat with all the batters, ending up with a stack of dill crepes and a stack of carrot crepes.
Heat 1 tablespoon oil in a separate skillet over medium-low heat. Cook onion, while stirring, until soft and translucent, about 5 minutes. Add mushrooms; cook until liquid evaporates, about 5 minutes more. Remove from heat and season with salt and pepper.
Blend cream cheese and 1/2 cup mayonnaise together in a bowl. Season with salt and pepper; add garlic and 2 sprigs dill.
Spread a portion of cream cheese mixture over a dill crepe; add a layer of the mushroom-onion saute. Top with a carrot crepe, spread with more cream cheese, and top with some of the chicken. Repeat layers until all crepes are used up. Decorate with thinly sliced cucumber. Cover with plastic wrap and chill until ready to serve.