Yes, this is definitely possible to do! :)
In order to avoid running into strange DNS lookup issues, you will want to ensure that the configuration is consistent across the different name servers. It means that your on-prem name servers should have the exact same records as the Route 53 name servers since a DNS lookup request may end up being handled by any one of these.
So the first step is to put all records you have on-prem into your Route 53 hosted zone as well. Also, Route 53 allows you to update the NS/SOA record sets to achieve your goal. For the NS records, you will want to have the combined list of name server addresses listed in both locations. The SOA record should match as well to ensure the configuration is fully consistent. Then it doesn't matter if a client ends up talking to an on-prem or Route 53 name server - they both have the same information.
As GoDaddy is your domain registrar, you then add the Route 53 name servers to the existing set of name servers for your domain through their management interface. In case you've been using GoDaddy as your DNS provider in the past, note that this is (likely) configuration done outside of the actual DNS record management. This step is about modifying the set of name servers for the domain itself.
We were able to take a soft cutover approach with zero downtime by adding the AWS NS at the bottom of the existing NS records in Godaddy. Gradually removing all old NS and adding rest of the AWS NS. AWS NS propagation was quicker than we expected. Query Logging is magical and a great tool to analyse the DNS traffic.
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