Set a wide kadai over medium-low heat. Add sesame seeds dry; stir often until fragrant and light gold with a few deeper flecks, about 4 to 7 minutes. Do not brown deeply or toast turns bitter.

Tip onto a plate and cool completely. Optionally reserve 2 tablespoons whole roasted til and pulse the remainder once or twice in a small grinder for a finer bind; mix cracked and whole together.

Wipe the kadai. Add jaggery and water. Heat on low until jaggery melts, stirring to break lumps. Raise heat slightly to bring to a gentle boil; skim foam if needed.

Boil on medium-low, stirring often, until syrup thickens and falls from the spoon in thin continuous threads.

Cold-water test: drip syrup into room-temperature water. When it forms a soft ball that holds shape yet still flattens slightly when pressed, you are in range. If it dissolves, keep cooking; if it snaps hard, you went too far (next batch pull earlier).

Turn heat to low.

Off heat or on lowest flame, stir in cardamom and optional peanuts and coconut, then add prepared til. Stir until coated and pulling together in a loose lump.

Rest until warm enough to handle but still pliable, about 1 to 2 minutes. Grease palms with remaining ghee. Shape rounds with a slight crack on top or smooth balls as you prefer.

Cool completely in a single layer before storing.