How to make custom annotation throw an error instead of default message

Question

I am working on an api that allows the user to fetch data by id, where the id's data type is a UUID. I would like to use an annotation in the controller method to check whether the user passed a valid UUID, but @Valid doesn't work on UUID's, so I want to write my own. Here is how I want to use my custom annotation in the controller:

@GetMapping("{id}")
    public ResponseEntity<MyObject> getById(@PathVariable @ValidUUID String id) {
        MyObject payload = service.getById(id);

        return ResponseEntity.status(HttpStatus.OK)
                .contentType(MediaType.APPLICATION_JSON)
                .body(payload);
    }

I have written a global exception handler:

import com.testapp.exceptions.InvalidUUIDException;
import org.springframework.http.HttpStatus;
import org.springframework.http.ResponseEntity;
import org.springframework.web.bind.annotation.ExceptionHandler;
import org.springframework.web.bind.annotation.RestControllerAdvice;

import java.time.LocalDateTime;
import java.util.LinkedHashMap;
import java.util.Map;

@RestControllerAdvice
public class GlobalExceptionHandler {

    @ExceptionHandler(InvalidUUIDException.class)
    public ResponseEntity<Object> handleInvalidUUIDException(InvalidUUIDException ex) {
        Map<String, Object> body = new LinkedHashMap<>();
        body.put("timestamp", LocalDateTime.now());
        body.put("message", ex.getMessage());

        return new ResponseEntity<>(body, HttpStatus.BAD_REQUEST);
    }
}

and a custom exception:

public class InvalidUUIDException extends RuntimeException {
    public InvalidUUIDException(String id) {
        super(String.format("The UUID provided (%s) is not valid", id));
    }
}

My problem is when I try to write the custom annotation, I can't figure out how to make it throw a custom exception. I found several discussions online, including this stack overflow question that talks about creating a custom annotation (the code below is from that question). All of the examples I have found just return a string, and don't allow throwing a custom exception (in my case InvalidUUIDException). Any thoughts on how to do this would be greatly appreciated.

@Target(ElementType.FIELD)
@Constraint(validatedBy={})
@Retention(RUNTIME)
@Pattern(regexp="^[0-9a-f]{8}-[0-9a-f]{4}-[1-5][0-9a-f]{3}-[89ab][0-9a-f]{3}-[0-9a-f]{12}$")
@ReportAsSingleViolation
public @interface UUID {
    String message() default "{invalid.uuid}";
    Class<?>[] groups() default {};
    Class<? extends Payload>[] payload() default {};
}