Blend flour, water, milk, eggs, 4 teaspoons vegetable oil, salt, and sugar together in a mixing bowl using an electric mixer until smooth. Cover and refrigerate for 1 hour.
Heat a crepe pan or skillet over medium heat. Brush with remaining 1 teaspoon oil. Stir batter well before pouring about 5 tablespoons into the heated pan to form a very thin coat. Cook crepe until set and bubbles start to appear on the surface, 1 to 2 minutes. Run a spatula around the edge of the skillet to loosen crepe; flip crepe and cook until the other side has turned light brown, about 1 minute more. Transfer crepe onto waxed paper. Repeat with remaining batter, being careful not to stack crepes until completely cooled.
Heat a saucepan over medium-high heat. Pour in orange juice and sugar. Bring to a boil; add mangoes, cinnamon, and vanilla extract. Return to a boil and cook for 2 minutes; set aside to cool.
Lay crepes on a flat, clean surface. Add mango filling to each and roll like a cigar. Place on a dessert plate; top with vanilla ice cream and chocolate syrup.