grillen
to grill
To cook food, especially meat or vegetables, on a grill over open embers or heat.
Am Wochenende wollen wir im Garten grillen.
At the weekend, we want to grill in the garden.

to grill / to brood
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to grill
To cook food, especially meat or vegetables, on a grill over open embers or heat.
Am Wochenende wollen wir im Garten grillen.
At the weekend, we want to grill in the garden.

The word 'grillen' has two very different meanings. Most of the time, it refers to preparing food on a grill. More rarely and colloquially, it means to think intensely and worriedly about something.
Lass uns am Wochenende am See grillen.
Let's grill by the lake this weekend.
When 'grillen' describes the social activity, it is often used without an object. The focus is then on the event itself, not on what exactly is being grilled.
Kommt ihr heute Abend zum Grillen vorbei?
Are you coming over for a barbecue this evening?
Common combinations are 'im Garten grillen' (to grill in the garden), 'auf dem Balkon grillen' (to grill on the balcony), or directly with the food: 'Würstchen grillen' (to grill sausages), 'Gemüse grillen' (to grill vegetables), or 'Steaks grillen' (to grill steaks).
Wir haben gestern Abend leckeres Gemüse auf dem Balkon gegrillt.
We grilled delicious vegetables on the balcony yesterday evening.
The meaning 'intensely think' is colloquial. It is mainly used in informal conversations among friends or family, often with a slightly negative or worried undertone.
Was grillst du schon wieder? Mach dir nicht so viele Sorgen.
What are you brooding over again? Don't worry so much.
to think
'nachdenken' and 'grillen' can both mean that you are mentally occupied with something in an intense way. 'nachdenken' is the normal word and fits in almost all situations, especially when you are calmly thinking about a question or problem. 'grillen' is colloquial here and tends to mean strenuous, worried brooding. Learners may confuse the words because both have to do with thoughts. The important point is: for neutral thinking something over, you say 'nachdenken'. 'grillen' sounds much more casual and only fits in certain colloquial situations; besides that, 'grillen' usually means preparing food on a grill.

Grill nicht die ganze Nacht wegen dieser einen E-Mail, morgen klären wir das.
Don't brood all night over that one email; we'll sort it out tomorrow.
Vor der Entscheidung denkt mein Bruder lange über den neuen Job nach.
Before making the decision, my brother thinks about the new job for a long time.