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Is there a way to re-deploy a resource whenever a dependency is changed?
for example: my API gateway stage resource needs to re-deploy every time when the resource or method is changed in the api-gateway

Not currently but we have an <https://github.com/pulumi/pulumi/issues/11577|issue> tracking it, and <https://github.com/pulumi/pulumi/pull/11089|prototype branch> to support it. If that looks like what your after I'd recommend adding adding a :+1:  to the issue to vote for it. Votes help with prioritisation.