I agree with Bacony. I think this is a combination of boredom/habit (just keep offering boring plain food) and that you're just not giving them enough calories.
The food you describe is great for an adult who is a healthy weight and size and watching what they eat to maintain that.
Children need more calories than you are describing. They are growing (huuuuuge amount of calories required to just sit in a chair and do nothing), and are active too. I'm not suggesting you feed them crap; I'm suggesting that you feed them foods like avocados, chickpeas (falafel, hummus), nut spreads (I see you mention a peanut allergy, are other nuts ok?), whole wheat seeded bagels, proper butter on their bread, corn-rich foods, fibre rich foods, coconut milk soups etc. Also, let them decide how much they eat. Your portions may be too small for them. Barring any abnormalities (which it sounds like your kids don't have), they will self regulate. The fact they are BOTH saying they are hungry suggests to me the issue is with supply, not demand.
Another tactic is to make sure that whatever you give at mealtimes is healthy (sounds like no problem there), and allow them to eat as much as they want at mealtimes. Then when they come looking for a snack say "sorry, you have to wait until [next meal] time, you should have had more at the table just now", and mean it and stick with it.
Finally, make sure they are not bolting their food. They should chew properly and eat not too fast so they get a chance to see if they really are full before they get down from the table.
I TOTALLY sympathies with not having time to do other stuff if they're constantly eating. In our house (I know it's MN sacrilege), no snacks allowed between meals apart from "teatime" (4/4.30, lunch being at 1pm and dinner at 6.30), when something small (sweet, savory, I really don't mind what) is allowed. Life is too short. As children nutrition shouldn't be a leisure activity. That's the sole preserve of food critics and weary adults 