4.5 months is too early for her immature and open GI tract. You will cause inflammation, irritation and introduce foreign bacteria. I swear I just answered this question the other day.... babies also lack a gag reflex and can aspirate food if solids is given too early.
Rice is not a good thing to give to an infant - even when they gut is ready for solids. Rice is binding and will cause stomach cramping, constipation and gas. Oatmeal is a much better thing to start off with.
I'm very glad to hear you are breastfeeding - and at 4, almost 5 months - that is ALL she should be eating. Babies go thru many growth spurts and that will cause them to eat a lot in shorter time frames. It's called cluster feeding. Don't think that this means you aren't producing enough, it just means the baby is priming your body for the next stage and in upping you production.
If you use that time to give her wasted (no nutrition) calories to fill her up, you will sabotage your milk supply. Just because a baby enjoys watching you eat and mimics you... does not mean they are developmentally ready to eat solids.