First of You should never post private information such as your public key of such API
Just by using the amazing Postman Chrome app you can see the result of such request:
click here to see the image below in full resolution

and I'm sure, if you instead of return client.Execute(request);
you do
var result = client.Execute(request);
return result;
and adding a breakpoint in the return
you can inspect what is the object that is passed from the call... without testing, I'm sure you can convert result.Content
(as it's where RestSharp appends the response content) into an object and use that object (or use the dynamic
type).
now, testing your code in VS:
click here to see the image below in full resolution

you can then use the dynamic
object like:
click here to see the image below in full resolution

public void GetResponse()
{
var client = new RestClient();
client.BaseUrl = "https://api.mailgun.net/v2";
client.Authenticator = new HttpBasicAuthenticator("api", "pubkey-e82c8201c292691ad889ace3434df6cb");
var request = new RestRequest();
request.Resource = "/address/validate";
request.AddParameter("address", "me@mydomain.com");
var response = client.Execute(request);
dynamic content = Json.Decode(response.Content);
bool isValid = content.is_valid;
string domain = content.parts.domain;
}
and treat the content of the response just like the json passed:
{
"address": "me@mydomain.com",
"did_you_mean": null,
"is_valid": true,
"parts": {
"display_name": null,
"domain": "mydomain.com",
"local_part": "me"
}
}