What's actually happening
Your smoke detector isn't broken most of the time. It's doing its job on air that can't escape.
In summer, cooking smoke drifts out an open window or door within seconds. In winter your house is buttoned up tight, so that same haze pools near the ceiling and rolls right into the detector's sensor.
The alarm keeps sounding because there are still particles in the chamber. Until you flush that air out, the sensor keeps reading them.
That's the short version. The longer version depends on where your detector sits, how old it is, and what kind of sensor it uses.
Why it keeps happening in the cold months
Start with airflow, because that's the culprit in most winter cases. A closed-up house recirculates its own air. Cooking particles that would've cleared in a minute now linger for 10 or 15.
Next, look at placement. Building codes generally want smoke alarms at least 10 feet from a cooking appliance. Plenty of older homes have one mounted right at the kitchen doorway, or in a hallway that funnels kitchen air straight at it.
Then there's the sensor itself. Ionization detectors react hard to the fine particles that normal cooking gives off. If yours is ionization and it's near the kitchen, that pairing is a nuisance-alarm machine.
Grease and dust matter too. Kitchen air is greasy air, and over months a film builds inside the detector. That buildup makes the sensor twitchy.
Finally, age. A detector past 10 years drifts. Old sensors false-trigger more, and they're the ones that won't reset cleanly after cooking.