Domatesleri kestikten sonra üzerlerine biraz zeytinyağı döktüm.

Because it is the direct object of kesmek inside kestikten sonra, and it refers to a specific set of tomatoes.

In Turkish, specific direct objects usually take the accusative ending. So:

• domates = tomato / tomatoes in a general sense
• domatesler = tomatoes
• domatesleri = the tomatoes / specific tomatoes

A useful extra note: domatesleri can also mean their tomatoes in other contexts. Here, though, it clearly means the tomatoes.

It means after cutting or after I cut.

It is built from kesmek (to cut):

• kes- = verb stem
• -tik / -dik = a verbal form used in structures like this
• -ten / -tan = ablative ending
• sonra = after

So:

• kesmek
• kestikten sonra = after cutting / after having cut

Because kes- ends in a voiceless consonant sound, the suffix appears as -tik rather than -dik.

In Turkish, with a structure like -Dıktan sonra, the subject is usually understood from context, and very often it is the same as the subject of the main verb.

Here the main verb is döktüm, which means I poured. So the natural reading is:

• After I cut the tomatoes, I poured...

If Turkish wanted to make a different subject clear, it would normally say so explicitly.

It means onto them or over them.

Here them refers to the tomatoes. Turkish often uses üzerine / üzerine dökmek for the idea of pouring something onto a surface.

So:

• domateslerin üzerine = onto the tomatoes
• üzerlerine = onto them

In this sentence, üzerlerine avoids repeating domateslerin.

Because Turkish often uses the dative for the destination or target of an action.

With dökmek (to pour), the thing you pour something onto commonly takes this kind of ending:

• masanın üzerine su dökmek = to pour water onto the table
• salatanın üzerine sos dökmek = to pour sauce over the salad

So üzerlerine is behaving like onto them rather than just on them.

Because biraz means a little / some, and after words like that, Turkish usually uses a bare singular-form noun.

So:

• biraz su = a little water
• biraz tuz = a little salt
• biraz zeytinyağı = a little olive oil

There is no plural because olive oil is a mass noun here, not something being counted.

There is also no accusative because this is an indefinite amount, not a clearly delimited, specific object.

Because it is a compound noun in Turkish.

It comes from:

• zeytin = olive
• yağ = oil

Together, zeytinyağı means olive oil. Turkish writes many common compounds as a single word.

Döktüm is the simple past, first person singular of dökmek (to pour / to spill).

So:

• döktüm = I poured

The -m at the end marks first person singular, so Turkish does not need to say ben unless it wants extra emphasis.

Not completely, but the given order is very natural.

Turkish usually puts the main verb at the end, so döktüm comes last. It is also common to put the time expression first:

• Domatesleri kestikten sonra... = After cutting the tomatoes...

You could also hear slightly different orders for emphasis, such as:

• Domatesleri kestikten sonra biraz zeytinyağı üzerlerine döktüm.

The meaning stays basically the same, but the focus shifts a little.