Translation guide
A fried egg is a common breakfast food. In Japanese, the most natural way to refer to it depends on how it's cooked and the context. The most common term is 目玉焼き (sunny-side up), but other terms exist for different styles.
fried egg (sunny-side up)
A fried egg cooked on one side with the yolk intact and runny, not flipped.
Literally 'eyeball fry', this is the standard word for a sunny-side up fried egg. It's the most common way to refer to a fried egg in Japanese.
I made a fried egg for breakfast.
I like my fried eggs sunny-side up with a runny yolk.
Literally 'fried egg', but this usually refers to a rolled omelette (tamagoyaki). In some contexts, it can mean a simple fried egg, but 目玉焼き is clearer.
卵焼き often implies the rolled Japanese omelette, not a Western-style fried egg. Use 目玉焼き to avoid confusion.
Please make a fried egg. (Could be interpreted as tamagoyaki.)
A fried egg flipped and cooked on both sides, with varying yolk doneness.
Referring to a fried egg used in a dish like a sandwich, rice bowl, or as a topping.
Still the most natural term when a fried egg is part of a dish.
I'll have a hamburger with a fried egg.
The direct loanword フライドエッグ exists but is rarely used in natural Japanese. It sounds like a menu item at a Western-style restaurant. In everyday conversation, always use 目玉焼き.
To specify how you want your fried egg cooked, you can add 半熟 (runny yolk) or 固焼き (well-done). For example: 半熟の目玉焼き (sunny-side up with runny yolk), 固焼きの目玉焼き (fried egg with firm yolk).
I want a fried egg cooked on both sides.
A fried egg rice bowl is easy and delicious.