Original code:
global class LightningForgotPasswordController {

    public LightningForgotPasswordController() {

    }

    @AuraEnabled
    public static String forgotPassword(String username, String checkEmailUrl) {
        try {
            Site.forgotPassword(username);
            ApexPages.PageReference checkEmailRef = new PageReference(checkEmailUrl);
            if(!Site.isValidUsername(username)) {
                return Label.Site.invalid_email;
            }
            aura.redirect(checkEmailRef);
            return null;
        }
        catch (Exception ex) {
            return ex.getMessage();
        }
    }

    @AuraEnabled
    global static String setExperienceId(String expId) {    
        // Return null if there is no error, else it will return the error message 
        try {
            if (expId != null) {
                Site.setExperienceId(expId);               
            }
            return null; 
        } catch (Exception ex) {
            return ex.getMessage();            
        }        
    } 
}

Test class:
@IsTest(SeeAllData = true)
public with sharing class LightningForgotPasswordControllerTest {

 /* Verifies that ForgotPasswordController handles invalid usernames appropriately */
 @IsTest
 static void testLightningForgotPasswordControllerInvalidUserName() {
  System.assertEquals(LightningForgotPasswordController.forgotPassword('fakeUser', 'http://a.com'), Label.Site.invalid_email);
  System.assertEquals(LightningForgotPasswordController.forgotPassword(null, 'http://a.com'), Label.Site.invalid_email);
  System.assertEquals(LightningForgotPasswordController.forgotPassword('a', '/home/home.jsp'), Label.Site.invalid_email);
 }

 /* Verifies that null checkEmailRef url throws proper exception. */
 @IsTest
 static void testLightningForgotPasswordControllerWithNullCheckEmailRef() {
  System.assertEquals(LightningForgotPasswordController.forgotPassword('a', null), 'Argument 1 cannot be null');
  System.assertEquals(LightningForgotPasswordController.forgotPassword('a@salesforce.com', null), 'Argument 1 cannot be null');
 }

 /* Verifies that LightningForgotPasswordController object is instantiated correctly. */
 @IsTest
 static void LightningForgotPasswordControllerInstantiation() {
  LightningForgotPasswordController controller = new LightningForgotPasswordController();
  System.assertNotEquals(controller, null);
 }
}