Place chickpea flour in a small bowl. In a thin stream, pour water into chickpea flour while whisking until batter is combined. Stir in 1 tablespoon olive oil and season with salt and pepper.
Heat a large nonstick skillet over medium. Lightly oil skillet. Pour a quarter of the batter into pan and cook until the pancake is browned on the edges and no longer wet-looking on top, about 3 minutes. Repeat with remaining batter. Transfer pancakes to cutting board, fold in half, and cut each pancake into two pieces.
While pancakes cook, finely chop olives and garlic. Stir olive mixture into butter with anchovy paste. Spread cut sides of bread with butter mixture, and top with chickpea pancake and parsley leaves.
Before her diagnosis, Caroline wrote a book on cakes called Cake Magic!. She started developing a birthday cake using her gluten-free mix found in that book. Check out other recipes she’s developing for her new life—and the stories behind them—on her blog, The Wright Recipes. Her next book, Catalan Food, written with chef Daniel Olivella, comes out in early September from Clarkson Potter.