EDIT May 2022:
As of May 2022, all SDKs have been re-released with native support for azure-identity. If you still encounter this error with a given SDK on its latest version, please open an issue asking for a re-release of that SDK here: https://github.com/Azure/azure-sdk-for-python/issues
Original response:
This is addressed here: https://learn.microsoft.com/en-us/azure/developer/python/azure-sdk-authenticate?tabs=cmd
Search "Using DefaultAzureCredential with SDK management libraries" on this page and it will take you to the section that covers your problem in more detail.
In a nutshell....
They updated the DefaultAzureCredential class and it no longer has a 'signed_session' attribute.  The newest versions of the management libraries should be updated to handle this.  As mentioned in another solution, update your azure-cli library to ensure you have the latest.  However, not all of the management libraries have been updated yet.  You can use this wrapper created by a member of the Azure SDK engineering team for the time being.
https://gist.github.com/lmazuel/cc683d82ea1d7b40208de7c9fc8de59d
# Wrap credentials from azure-identity to be compatible with SDK that needs msrestazure or azure.common.credentials
# Need msrest >= 0.6.0
# See also https://pypi.org/project/azure-identity/
from msrest.authentication import BasicTokenAuthentication
from azure.core.pipeline.policies import BearerTokenCredentialPolicy
from azure.core.pipeline import PipelineRequest, PipelineContext
from azure.core.pipeline.transport import HttpRequest
from azure.identity import DefaultAzureCredential
class CredentialWrapper(BasicTokenAuthentication):
    def __init__(self, credential=None, resource_id="https://management.azure.com/.default", **kwargs):
        """Wrap any azure-identity credential to work with SDK that needs azure.common.credentials/msrestazure.
        Default resource is ARM (syntax of endpoint v2)
        :param credential: Any azure-identity credential (DefaultAzureCredential by default)
        :param str resource_id: The scope to use to get the token (default ARM)
        """
        super(CredentialWrapper, self).__init__(None)
        if credential is None:
            credential = DefaultAzureCredential()
        self._policy = BearerTokenCredentialPolicy(credential, resource_id, **kwargs)
    def _make_request(self):
        return PipelineRequest(
            HttpRequest(
                "CredentialWrapper",
                "https://fakeurl"
            ),
            PipelineContext(None)
        )
    def set_token(self):
        """Ask the azure-core BearerTokenCredentialPolicy policy to get a token.
        Using the policy gives us for free the caching system of azure-core.
        We could make this code simpler by using private method, but by definition
        I can't assure they will be there forever, so mocking a fake call to the policy
        to extract the token, using 100% public API."""
        request = self._make_request()
        self._policy.on_request(request)
        # Read Authorization, and get the second part after Bearer
        token = request.http_request.headers["Authorization"].split(" ", 1)[1]
        self.token = {"access_token": token}
    def signed_session(self, session=None):
        self.set_token()
        return super(CredentialWrapper, self).signed_session(session)
if __name__ == "__main__":
    import os
    credentials = CredentialWrapper()
    subscription_id = os.environ.get("AZURE_SUBSCRIPTION_ID", "<subscription_id>")
    from azure.mgmt.resource import ResourceManagementClient
    client = ResourceManagementClient(credentials, subscription_id)
    for rg in client.resource_groups.list():
        print(rg.name)
Your code would then look like:
from cred_wrapper import CredentialWrapper
credentials = CredentialWrapper()
compute_client = ComputeManagementClient(credentials, subscription_id)
# Starting the VM
print('\nStarting VM ' + VM_NAME)
vm_start = compute_client.virtual_machines.start(
    RG_NAME, VM_NAME)
vm_start.wait()
You should be cooking from there!