Meine Freundin lebt fast vegan und macht aus Aubergine, Lauch und Süßkartoffel ein warmes Gericht.

Because Freundin is a feminine singular noun and it is the subject of the sentence, so it is in the nominative case. The possessive mein- has to agree with that, so you get meine Freundin.

Compare:

• mein Freund = my boyfriend / male friend
• meine Freundin = my girlfriend / female friend

In German, all nouns are capitalized, not just proper names. So Freundin, Aubergine, Lauch, Süßkartoffel, and Gericht all start with capital letters.

That is a normal spelling rule in German.

German often uses vegan leben or vegetarisch leben to mean to live vegan or to live as a vegan. It refers to someone's lifestyle or usual way of eating.

So:

• vegan leben = to live vegan
• vegetarisch leben = to live vegetarian

In English, live vegan is possible, but German uses this structure very naturally.

No. Fast is a false friend here. In German, fast usually means almost or nearly.

So fast vegan means:

• almost vegan
• more or less vegan
• not completely vegan, but very close

If you want the English meaning fast = quick, German usually uses schnell.

Because in a coordinated sentence with und, German often leaves out the second subject if it is the same as the first one.

So this:

• Meine Freundin lebt fast vegan und macht ...

really means:

• Meine Freundin lebt fast vegan und meine Freundin macht ...

But repeating meine Freundin would sound unnecessary. You could also say und sie macht ..., but leaving it out is very common and natural.

The pattern aus etwas etwas machen means to make something out of something.

So:

• aus Aubergine, Lauch und Süßkartoffel ein warmes Gericht machen = to make a warm dish out of aubergine, leek, and sweet potato

Using macht focuses on the result: the ingredients are turned into a dish.

Kocht would emphasize the act of cooking:

• Sie kocht ein warmes Gericht. = She cooks a warm dish.

Both can work, but macht aus ... highlights what it is made from.

Aus takes the dative case.

That means the ingredients after aus are logically dative:

• aus Aubergine, Lauch und Süßkartoffel

In this sentence, you do not see special case endings because there are no articles and these noun forms do not change here.

If articles were added, the dative would become visible:

• aus der Aubergine
• aus dem Lauch
• aus der Süßkartoffel

German often omits articles when listing ingredients, especially in a general or recipe-like style.

So aus Aubergine, Lauch und Süßkartoffel sounds like:

• made from aubergine, leek, and sweet potato

This is similar to English ingredient lists, where you also often leave out articles.

If you wanted to be more specific, you could add articles or quantities:

• aus einer Aubergine, einem Lauch und einer Süßkartoffel
• aus der Aubergine, dem Lauch und der Süßkartoffel
  if you mean specific known items

In ingredient language, German often uses the singular to name an ingredient as a type of food, not necessarily to count exact individual items.

So Aubergine, Lauch und Süßkartoffel can sound like:

• aubergine, leek, and sweet potato as ingredients

It may suggest one of each, or just those ingredients in general.

A plural is also possible in other contexts, for example:

• aus Auberginen und Süßkartoffeln

That would sound more like several aubergines and several sweet potatoes, or ingredients in a more general plural sense.

Because Gericht is a neuter noun, and here it is in the accusative singular as the direct object of macht.

With ein + neuter accusative singular, the adjective takes -es:

• ein warmes Gericht

Compare:

• ein warmes Gericht = a warm dish
• ein kaltes Gericht = a cold dish

This is part of German adjective endings.

Because Gericht is neuter, not masculine.

So:

• masculine accusative: einen warmen ...
• neuter accusative: ein warmes ...

Compare:

• einen warmen Tee = a warm tea (masculine)
• ein warmes Gericht = a warm dish (neuter)

The article and adjective ending both depend on the gender of the noun.

The structure is:

• macht = finite verb
• aus Aubergine, Lauch und Süßkartoffel = prepositional phrase
• ein warmes Gericht = direct object

So literally it is:

• She makes, out of aubergine, leek and sweet potato, a warm dish.

German often places the aus-phrase before the direct object in this kind of sentence, especially with aus etwas etwas machen. It sounds natural because the pattern is very common:

• aus X Y machen = to make Y out of X

So the sentence is grammatically very normal as written.