How can I ask for more of an ingredient in a dish? like add extra cheese, extra meat, etc. Not that it matters but I specifically want to know this for 手抓餅.
If the ingredient you asking for more is free, like condiments (salt, sugar, oil) or side greens (onion, pickle) you can say "多 ~ ":
If the ingredient you asking for more will be charged money, like main topping (meat, seafood) or staple food (noodle, rice) Then you can say "加 ~" :
Some western restaurant offer free bread.
加鱼肉， 加牛肉， 加猪肉， 加面条， 加饭
Some restaurant have set price for extra ingredient and some don't, so you should ask how much it costs to have extra something.
To get a larger size dish, you can say 加大码, again, Some restaurant have set price for extra large size dishes and some don't, you have to ask for the the price.
I often order 例半 (one and a half order) if there are a lot of people in the party. Since 例半 is one and a half order, I would expect it would be one and a half price.
Here are some examples of how you order extra
牛肉炒麵，多洋蔥 (beef stir fry boodle, more onion)
牛肉炒麵，加牛肉 (beef stir fry boodle, extra beef) - ask the price or state how much more money you want to pay for it e.g. "加十元牛肉" (add ten dollar beef)
牛肉炒麵，加大码 (beef stir fry boodle, large size) ask the price
牛肉炒麵，例半 (beef stir fry boodle, one and a half order) pay 50% more
馄饨面，加底 (wonton noodle soup, extra noodle) - 加底 means add only extra staple food when the dish is made up of main toppings on stable food base, For example 馄饨面 (wonton noodle soup)
For 手抓餅, I would say: 老板，来一个手抓饼。加鸡蛋，加培根，加蔬菜.
The common verbs here are 加(add) and 放(put). E.g. 不要放培根；加火腿肠;