Zum Frühstück mache ich mir ein Rührei mit Lauch.

Zum is a contraction of zu dem.

In this sentence, zum Frühstück is an idiomatic way to say for breakfast or at breakfast time.

• zu dem Frühstück → zum Frühstück
• Similar expressions:
  • zum Mittagessen = for lunch
  • zum Abendessen = for dinner

It does not literally mean to the breakfast in normal English. It is just the standard German way to express this idea.

German main clauses follow the verb-second rule.

That means the finite verb must come in the second position, no matter what comes first.

Here, Zum Frühstück is placed first, so the verb mache must come next:

• Zum Frühstück | mache | ich ...

If you start with the subject instead, that is also correct:

• Ich mache mir zum Frühstück ein Rührei mit Lauch.

Both are grammatical. Starting with Zum Frühstück gives a little more emphasis to for breakfast.

Mir is the dative form of ich. Here it means something like for myself.

So:

• Ich mache ein Rührei. = I make scrambled eggs.
• Ich mache mir ein Rührei. = I make myself scrambled eggs / I make scrambled eggs for myself.

This is very common in German when someone prepares food, coffee, tea, etc. for themselves.

It is not absolutely required, but it adds the idea that the food is for your own benefit.

German treats Rührei as a singular noun meaning the dish scrambled eggs.

So:

• das Rührei = scrambled eggs
• ein Rührei = a serving/portion of scrambled eggs

Even though English uses a plural expression, German usually uses this singular noun for the dish as a whole.

Ein Rührei is in the accusative because it is the direct object of machen.

You are making what?
→ ein Rührei

So:

• ich = subject
• mache = verb
• ein Rührei = direct object

Because Rührei is neuter (das Rührei), the article ein looks the same in nominative and accusative:

• nominative: ein Rührei
• accusative: ein Rührei

So the case is accusative, even though the form does not change.

Mit always takes the dative case, but here there is no article, so you do not see a case ending on the noun.

• mit Lauch = with leek

This is very natural when naming ingredients in a general way. It works like an ingredient label:

• mit Käse
• mit Tomaten
• mit Lauch

If you said mit dem Lauch, that would usually mean a specific leek already known from the context.

Also note:

• mit
  • dative
• with an article, you would see it clearly: mit dem Lauch

In everyday German, machen is very common for preparing food.

So Ich mache mir ein Rührei sounds natural and conversational.

Compare:

• machen = make / prepare
• kochen = cook
• zubereiten = prepare (more formal or neutral)

For a simple meal, Germans often prefer machen.

So this sentence sounds like natural spoken German, not like a textbook-only phrasing.

Yes, absolutely.

This version is also correct:

• Ich mache mir zum Frühstück ein Rührei mit Lauch.

The difference is mainly focus:

• Zum Frühstück mache ich mir ...
  → emphasizes for breakfast
• Ich mache mir zum Frühstück ...
  → more neutral, starting with I

Both are normal German.

Yes. Mit Lauch describes ein Rührei.

So the structure is basically:

• Zum Frühstück = for breakfast
• mache ich mir = I make myself
• ein Rührei mit Lauch = scrambled eggs with leek

In other words, the dish you are making is scrambled eggs with leek.

Not quite.

• zum Frühstück = for breakfast / as breakfast
• beim Frühstück = during breakfast / while eating breakfast

So in your sentence, zum Frühstück is the right choice because it tells us what you are having or making for breakfast.

If you said beim Frühstück, it would sound more like something happening during breakfast time.