Heat the oil in a large skillet over medium-high heat. Add the mushrooms, 1/4 teaspoon of the salt, and 1/8 teaspoon of the pepper and cook for 3 minutes. Add the beef and cook until no trace of pink remains, 5 to 6 minutes. Spoon off and discard any fat. Add the brandy and cook, stirring, until almost evaporated, 1 to 2 minutes. Stir in the spinach and cook until heated through, about 1 1/2 minutes. Season with the remaining salt and pepper. Remove from heat and stir in the Parmesan.
Cut a 2-inch strip off the side of each pita and pull open. Spread the inside with the mayonnaise (if using). Fill each pita with the beef and mushroom mixture. Serve with the carrots.