Stir water and brown sugar together in a small bowl. Sprinkle yeast evenly over the surface. Cover with a clean cloth and let stand until yeast softens and forms a creamy foam, about 7 minutes.
Mix all-purpose flour, whole wheat flour, shortening, and salt together in a large bowl until shortening is thoroughly incorporated. Pour in yeast mixture; mix with your hands or a wooden spoon until dough comes together.
Dust a flat work surface with flour. Knead dough until smooth and holding its shape.
Lightly grease a large bowl with 1 teaspoon vegetable oil. Add dough; turn to coat with oil. Cover with a clean cloth and let rise until doubled, about 35 minutes.
Gently punch down dough to deflate. Dust your work surface with more flour. Roll dough out until it is 1/3-inch thick. Fold in half and roll out again. Repeat this process 4 times.
Fill a large baking pan with 1 inch water and place it on the bottom rack of the oven. Preheat oven to 350 degrees F (175 degrees C).
Lightly grease a jelly roll pan with 1 teaspoon vegetable oil. Cut dough into 9 pieces and place on the greased pan. Cover with a clean, damp cloth; let rise until puffy, about 25 minutes.
Bake in the preheated oven above the water until browned on top, 15 to 20 minutes.