Rinse the red lentils thoroughly under cold water until the water runs clear. This removes the surface starch that can make the final purée slightly gluey if skipped.
Place the rinsed lentils and chopped carrot into a small saucepan. Add the 300ml of water. Bring to the boil over medium heat, then reduce to a gentle simmer.
Cook uncovered for 15 to 18 minutes, stirring occasionally, until the lentils have completely broken down and the carrot is very soft. The lentils will absorb most of the water and the mixture will thicken considerably, this is exactly what you want. Remove and set aside.
In a saucepan, heat the olive oil and cook the chopped onions until golden brown. Add the cooked lentils and let simmer for 2 minutes before turning off the heat.
Remove from the heat and allow to cool for a few minutes. Transfer to a blender or use a hand blender to purée until completely smooth. Add a small amount of cooled boiled water, a teaspoon at a time, until you reach the consistency you need - very smooth and thin for a first food at six months, slightly thicker as your baby progresses through the early weeks of weaning.