Thinly slice the red and yellow onions and place them in a bowl. Let sit for 10 minutes to draw out excess moisture, which helps in crisping.
In a large mixing bowl, combine chickpea flour, turmeric, cumin, chili powder, and salt. Whisk together to evenly distribute the spices.
Gradually add water to the dry mixture, whisking until you get a thick, pancake-like batter that coats the onions well. Adjust with more water or chickpea flour if needed.
Drain the sliced onions from excess moisture and fold them into the batter, ensuring each piece is well coated with the spicy chickpea mixture.
Pour about 2 inches of oil into your deep-fryer or heavy-bottomed pot and heat to 180°C (356°F). Test the temperature by dropping a small spoonful of batter—if it sizzles and rises steadily, you're ready to fry.
Carefully drop spoonfuls of the onion batter into the hot oil, a few at a time. Fry for 3-4 minutes, turning occasionally, until the bhajis are golden brown and crispy on all sides, with a crackling sound.
Use a slotted spoon to lift the fried bhajis out of the oil and place them on paper towels to drain excess oil. Rest for 2 minutes to let the crust set and cool slightly.
Squeeze fresh lemon over the crispy bhajis and serve immediately. The aroma should be smoky and spicy, with a satisfying crunch and tender onion inside.