Did you do a PUT and actually send some JSON in the body as per the example?
curl -H 'Authorization: Bearer eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJleHAiOjE1MDIyOTg3ODksInVzZXJfaWQiOiIyNzQxOTAxMSIsInJvbGUiOiJleHRlcm5hbCJ9.sKjspl1jw0XGcpjF8vo2IcS2dQLQU_rSsmRCnrohCP4' \
-H 'Client-Id: pxifeyz7vxk9v6yb202nq4cwsnsp1t' \
-H 'Content-Type: application/json' \
-d '{"required_configuration": "RCS-1"}' \
-X PUT https://api.twitch.tv/extensions/pxifeyz7vxk9v6yb202nq4cwsnsp1t/0.0.1/required_configuration?channel_id=27419011
Looking at your link I see you payload is bad