[EKS/Fargate] [request]: Have Fargate pods pull search domains from VPC DHCP Options Sets

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**Tell us about your request**
Have Fargate pods running on EKS pull search domains from, VPC DHCP Options Sets

**Which service(s) is this request for?**
EKS/Fargate

**Tell us about the problem you're trying to solve. What are you trying to do, and why is it hard?**
On our VPC, we have custom domain names set on our DHCP Options Sets which allow us to query internal domain names on custom R53 private hosted zones linked to the VPC without having to specify the full domain name. For example, our VPC DHCP Options Sets is set to `domain-name = confidential.internal us-west-2.compute.internal`, and instances on the VPC can query any internal hostname under the `confidential.internal` hosted zone without having to specify the domain.

Spinning up a pod on an EKS EC2 worker node recognizes this just fine, and adds the custom domain name to it's `/etc/resolv.conf` under the search domain, as so:
```
# cat /etc/resolv.conf
nameserver 10.100.0.10
search dev.svc.cluster.local svc.cluster.local cluster.local confidential.internal. us-west-2.compute.internal
options ndots:5
```

Spinning up the same pod on a Fargate worker node in the same cluster (same VPC and subnet, of course), doesn't seem to recognize this, with the `/etc/resolv.conf` being:
```
# cat /etc/resolv.conf
search dev.svc.cluster.local svc.cluster.local cluster.local us-west-2.compute.internal
nameserver 10.100.0.10
options ndots:5
```

For consistency's sake, it would be great if the Fargate pods would be able to pull these domain names from DHCP Options Sets, and apply it to `/etc/resolv.conf` as expected.

**Are you currently working around this issue?**
Workaround now is to add the custom domain name to the search portion `/etc/resolv.conf` file upon building the container, or explicitly specify all lookups to the custom hosted zone with the full domain name.

**Additional context**
We've tried the same case previously on ECS Fargate nodes, and those seem to work just fine.

No repository files, tests, or entry points are named. Start by investigating how EKS/Fargate constructs pod /etc/resolv.conf and how VPC DHCP Options Sets are exposed, comparing the issue's EC2 and Fargate examples. Done means Fargate pods include the configured custom search domains while retaining the existing Kubernetes search domains.

Written by the indexing model from the issue text.