cypress-spy-emit.ts

cy.get('.button').click().then(() => {
  cy.wrap(Cypress.vueWrapper.emitted()).should('have.property', 'click-forgot-password-link')
})

Vuetify's next branch has a bunch of Cypress component testing examples. Button. They use TSX but you can do something like this:

const onRegister = cy.stub()

mount(KongAuthRegister, {
  props: { 
    onRegisterSuccess: onRegister
  }
})

cy.intercept('POST', '**/register', {
  body: {
    organizationID: '187e2b65-ec69-421c-a7ba-3e946c4e5077',
  },
}).as('register-request')

// stuff ...

cy.wait('@register-request').its('response.body').should('have.property', 'organizationID').then(() => {
  // Check for emitted event
  expect(onRegister).to.have.been.calledWith(......) // <================== your object here??
})

The trick is Vue turns @register-success into an onRegisterSuccess props when it compiles.

Example: lmiller1990/vue-3-cypress-vite#2