Patlıcanları fırına vermeden önce üzerlerine biraz salça sürdüm.

Because it contains two things:

• patlıcan = eggplant / aubergine
• -lar = plural
• -ı = accusative case

So patlıcanları means the eggplants as a specific direct object.

In Turkish, a specific/definite direct object usually takes the accusative. Compare:

• patlıcan kestim = I cut eggplant / some eggplants
• patlıcanları kestim = I cut the eggplants

Because fırına uses the dative ending -a/-e, which often shows movement toward or into something.

• fırın = oven
• fırına = to the oven / into the oven

So fırına vermek means to put into the oven.

By contrast:

• fırında = in the oven
• fırını = the oven (as a specific direct object)

It is a very common Turkish pattern:

• verb stem + -meden/-madan önce = before doing X

So:

• vermek = to give / put
• vermeden önce = before giving / before putting

In this sentence, it means before putting them in the oven.

A useful comparison:

• fırına vermeden önce = before putting into the oven
• fırına verdikten sonra = after putting into the oven

Yes, vermek often means to give, but in cooking it is also used idiomatically for putting something into heat, especially an oven.

So fırına vermek is a natural Turkish expression meaning:

• to put in the oven
• to bake

It is not a literal English-style give to the oven. It is just the standard Turkish way of saying it.

Üzerlerine means onto them, on top of them, or over them.

It refers back to patlıcanları. So the idea is:

• üzerine = onto it
• üzerlerine = onto them

In this sentence, it means the tomato paste was spread on the eggplants.

A very close alternative would be:

• patlıcanların üzerine biraz salça sürdüm

That means almost the same thing.

Because biraz salça is an indefinite, nonspecific object: some tomato paste.

In Turkish, indefinite direct objects often appear without the accusative ending.

So:

• biraz salça sürdüm = I spread some tomato paste

If you said salçayı sürdüm, it would sound more like:

• I spread the tomato paste
• a specific, known paste

Here sürmek means to spread, to smear, or to apply onto a surface.

That is a very common use:

• krem sürmek = to apply cream
• tereyağı sürmek = to spread butter
• salça sürmek = to spread tomato paste

So in this sentence, sürdüm means I spread/applied.

It is built into the verb sürdüm.

• sür-dü-m
• -m = I
• -dü- = past tense marker

So sürdüm means I spread.

Turkish often leaves out subject pronouns like ben when the verb ending already shows who the subject is.

Turkish normally puts the verb at the end, and other parts come before it.

So this sentence follows a very natural pattern:

• Patlıcanları = the eggplants
• fırına vermeden önce = before putting them in the oven
• üzerlerine = onto them
• biraz salça = some tomato paste
• sürdüm = I spread

A literal structure would be something like:

• The eggplants, before putting in the oven, onto them some tomato paste I spread.

That sounds odd in English, but it is normal in Turkish.

Yes. That is also natural.

The difference is mainly stylistic:

• üzerlerine = onto them, referring back to the eggplants
• patlıcanların üzerine = onto the eggplants, stating it explicitly again

Since patlıcanları has already been mentioned, üzerlerine avoids repeating the noun and sounds very natural.

It is the simple past (also called the definite past in Turkish grammar).

• sürmek = to spread
• sürdüm = I spread / I did spread

It describes a completed action in the past.

So the whole sentence tells us about something the speaker already did: first they spread the paste, then the eggplants went into the oven.