public class BaseController : ApiController {
            [HttpPost]
            public IHttpActionResult Post(T entity)
            {
                return Ok();
            }

            [HttpPut]
            public IHttpActionResult Put(T entity)
            {
                return Ok();
            }
    }

public class ObjectController : BaseController {
    [HttpPut]
    [Route("api/Object")]
    public new IHttpActionResult Put(T entity)
    {
        return Ok();
    }
}