Now that my kids are older and have lunch at school (and vastly different lunchtimes at that!), I find that when we're home or out as a family, we let them snack when they're hungry and eat meals when they need to, and I sort of did that when they were younger too (even though mealtimes weren't as far apart as 11AM and 1PM!). If she's not properly hungry till 11AM, I'd go ahead and feed her a decent meal then and call it lunch and call the milk drink "breakfast." If she's having milk when she wakes, will she take anything with that, like a bite of banana, or is that enough for her? Natalie would drink a (rice) milk in the morning and her appetite was all over the place, so we ended up adding some protein powder to it, putting it in a sports bottle, and she'd toddle out of her room, find it waiting for her at the end of the hallway, shake it up, and guzzle it for breakfast. (I used that particular protein powder because she was in an anti-veg stage at the time and this had some veg greens and juice hidden in it.
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As it turns out, each of my girls has a very different set of food and meal preferences: they like different foods, and at different times, in different quantities - I have one who's more a "carnivore," and another who self-selects away from most red meats, and while we ask them to try different things, they also know by now (at 13 and almost-10) what makes them feel good and what they just don't care for, so I don't mind making meals that I can adapt for them (like maybe a soup or stew where Natalie doesn't have much if any of the meat but does have more grains and veg). I do think it's important to let them learn to really listen to their bodies rather than just eating mindlessly.
That's something I think we all learn gradually over time, given enough opportunity to do so.
