Después de remojar los garbanzos toda la noche, mi tía los pone a cocer por la mañana.

Because after después de, Spanish often uses an infinitive when talking about after doing something in a general way:

• después de remojar = after soaking
• después de comer = after eating
• después de llegar = after arriving

This is very common and natural.

If you want a full clause with a subject and a conjugated verb, you normally use después de que:

• Después de que remoja los garbanzos... is not the most natural choice here.
• More natural would be something like Después de que los garbanzos han estado en remojo toda la noche...

So in this sentence, después de remojar is the normal compact way to say after soaking.

Yes. Remojar means to soak or to leave in water so something softens.

In cooking, remojar los garbanzos and poner los garbanzos en remojo are both natural:

• remojar los garbanzos
• poner los garbanzos en remojo

The second one is a little more literally to put the chickpeas to soak / in soaking water, but both are common.

In this context, it usually means overnight.

Literally, toda la noche is all night / the whole night, but in cooking instructions it often means that something stays soaking for the entire night until the next morning. So the idea is not dramatic emphasis, just the normal cooking step of leaving chickpeas soaking overnight.

Los is a direct object pronoun that refers back to los garbanzos.

So:

• los garbanzos = the chickpeas
• los = them

Spanish often avoids repeating the full noun again when it is already clear:

• Después de remojar los garbanzos..., mi tía los pone a cocer.

This is very natural. English does the same thing:

• After soaking the chickpeas..., my aunt puts them on to cook.

Because object pronouns normally go before a conjugated verb:

• los pone = she puts them

So the structure is:

• mi tía = subject
• los = direct object pronoun
• pone = verb

This is standard Spanish word order.

You could also say the sentence without the pronoun by repeating the noun:

• mi tía pone los garbanzos a cocer

Both are correct, but los is more natural here because los garbanzos was just mentioned.

Poner a cocer means something like to put on to boil / to start cooking by boiling.

It is a very common cooking expression in Spanish. The pattern is:

• poner + object + a + infinitive

Here:

• poner los garbanzos a cocer = to put the chickpeas on to cook/boil

The idea is not just cook in a general sense, but specifically set them to boil or simmer.

Because cocer is more specific.

• cocinar = to cook in a general sense
• cocer = to cook in liquid, especially to boil or simmer

For chickpeas, cocer is the natural verb because they are cooked in water. So:

• poner a cocer los garbanzos sounds very natural
• poner a cocinar los garbanzos is much less idiomatic here

In Spain, cocer is especially common in this kind of cooking sentence.

In Spain, por la mañana is the usual way to say in the morning.

These are common time expressions:

• por la mañana = in the morning
• por la tarde = in the afternoon/evening
• por la noche = at night / in the evening

En la mañana does exist, but it is less typical in Spain and is more common in some parts of Latin America. For Spain-focused Spanish, por la mañana is the best choice.

Because the present tense in Spanish is often used for habitual actions, routines, and recipe-style descriptions.

So mi tía los pone a cocer por la mañana can mean:

• this is what she usually does
• this is her normal routine
• this is the next step in the cooking process

Spanish uses the present tense a lot in this way, just like English can say My aunt soaks them overnight and cooks them in the morning.

Not 100% grammatically explicit, because después de remojar does not state the subject directly.

In practice, most readers will naturally understand that my aunt is the person doing the soaking, or at least that this is part of her process. But the infinitive construction itself does not spell the subject out.

If you wanted to make the subject clearer, you could rephrase:

• Después de dejar los garbanzos en remojo toda la noche, mi tía los pone a cocer por la mañana.
• Después de que los garbanzos hayan estado en remojo toda la noche, mi tía los pone a cocer por la mañana.

So the original sentence is natural, but slightly less explicit about who performed the soaking.

Yes. Here are some natural alternatives:

• Después de poner los garbanzos en remojo toda la noche, mi tía los pone a cocer por la mañana.
• Tras remojar los garbanzos toda la noche, mi tía los pone a cocer por la mañana.
• Después de dejarlos en remojo toda la noche, mi tía los pone a cocer por la mañana.

A few notes:

• tras is a bit more formal or written than después de
• dejarlos en remojo is very common in cooking
• all of these keep basically the same meaning

It is more general. Poner a + infinitive can mean to set someone/something to do something or to begin doing something, depending on the context.

Examples:

• poner el agua a hervir = to put the water on to boil
• poner la carne a asar = to put the meat on to roast
• ponerse a estudiar = to start studying

In your sentence, poner los garbanzos a cocer is a cooking-specific version of that same pattern.