What is the best discipline for a 3 1/2 year old?

She is an all around good little girl, but we clash in one area- mealtimes. i do my best to not make a big deal, to fix something she likes with it, etc. etc. but she refuses to eat!
For example, today: made her one egg and a hashbrown for breakfast. She likes hashbrowns. refused to try the egg. Sat at the table for an hour and a half. i asked/ told her to take one bite. she refused. After almost 2 hours, i got her to try one bite- which she gagged on. Whatever, trid to play it off as she did a good job, etc....
lunchtime: made mac-n-cheese (which she likes). She sat at the table for 2 HOURS! Still did not finish the small bowl of it. Thinking I will just keep on schedule with my day, and not let her run it, I took her down in time for her nap. She did that ok.
got up, everything fine (except crabby due to not eating)

Tonight's the kicker though. Trying to try another technique... making her sit at the table without attention until she eats it (or at LEAST tries it!!) I made grilled chicken and rice. That was at 5:30.
right now, it's 7:46. So far she's spent 5 minutes repeating the time after our clock chimed, she's gone to the bathroom 2 times (after crying about it for 15 minutes the second time). She's had nothing to drink, as she fills up on it and says she's full.... She's doing everything she can to get my attention (short of actually getting down or throwing her food). Now she's pestering me that she's tired and needs to go to bed. i am still ignoring her. I'm not sure if this is the right thing to do or not. She is very strong-willed when it comes to meal times! She has gone a whole day without eating ANYTHING just because she doesn't want what i've made. I tried the re-heating a meal, I've tried making something she likes with it (she'll eat that). she'd rather take a nap, go in time-out, go without a snack/ desert/ fun, anything.
I made brownies for desert.. and she's trying to explain to me she needs one before she goes to bed or eats her dinner! LOL Oh, now it's medicine she needs..... LOL I know it's just for attention, but I don't understand why???? Anyone watch Supernanny?? hahaha
I'm totally not sure what to do in this situation!?!?! Should I let her fall asleep at the table? how do i teach her to stop fighting and just try a bite at mealtime and not be so dramatic??

Thanks everyone for your insight!
she also spends time with me in the kitchen, helps me make the meals etc... but when it comes time to eating, suddenly it's " I don't like that"!
Do I just give in and only make what she will eat if it's healthy? The only proteins she will eat is hot dogs, pepperoni off a pizza (not homemade though), peanutbutter, and chicken nuggets. Grrr... she will eat fresh veggies...(carrots and cucumbers and celery) and she eats alots of fruit. I'm tired of only making meals she likes or making her something different....
signed, frustrated mommy.
I repeat. she is NOT throwing food. all the drama is just her talking and crying.
 
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