Hi,
You may have to open it up and check what type of thermostat it has.
Some are just the bi-metal type where as they heat up the two metal strips move away from each other and open the circuit for the power to the heater elements. Turning the setting alters the amount of travel required i.e. the time it takes. (examples only to show what they might look like)
Others use an electronic timer circuit which is adjusted by the darkness setting control.
Hopefully if you open it up you'll be able to spot it easily and maybe with luck there will be some part information stamped on the thermostat to help find a replacement part.
Post some images of the control board, thermostat etc back here. Here's how to add images to your question in ifixit Adding images to an existing question
Be safety aware when working on toasters as there's exposed lethal AC voltage present if the power is connected. Always disconnect the power when working in a toaster unless you're electrically testing it and you know what you're doing.
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