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A simple component:

<template>
 <div>
 <p>
 count 
 </p>
 <button @click="increment" data-test="increment">Increment</button>
 </div>
</template>
<script>
 export default 
 data () 
 return 
 count: 0
 
 ,
 methods: 
 increment () 
 this.count++
 
 
 
</script>

And my test:

import TestExperiment from '@/components/TestExperiment'
import createLocalVue, shallowMount from '@vue/test-utils'

const localVue = createLocalVue()

describe('testexperiment.test.js', () => 
 const cmp = shallowMount(TestExperiment, 
 localVue
 )

 const increment = jest.spyOn(cmp.vm, 'increment')
 const incrementButton= cmp.find('[data-test="increment"]')

 test('clicking increment button calls increent', () => 

 expect(incrementButton.exists()).toBe(true)

 incrementButton.trigger('click')

 // Checking call here fails:
 // expect(increment).toHaveBeenCalled()

 // Function was still obviously called
 expect(cmp.vm.count).toBe(1)

 incrementButton.trigger('click')

 // Checking call here passes:
 expect(increment).toHaveBeenCalled()
 )

)

As you can see, I trigger a click on incrementButton twice.

After the first call, if I test if the method 'increment' was called, it returns false. However, count is indeed incremented. After the second call, it registers that it was in fact called (if I test how many times it was called, it asserts that it was called one time, even if count is 2, clearly having been incremented twice).

What am I missing about how Jest/Vue works?

You need to use the Vue Test Utils setMethod method:

const wrapper = mount(Foo)
const clickMethodStub = sinon.stub()

wrapper.setMethods( clickMethod: clickMethodStub )
wrapper.find('button').trigger('click')

expect(clickMethodStub.called).toBe(true)

The reason it does not work in your example is because the element that you dispatch a click on has the original handler that the component created when it was instantiated.

It works after two calls to trigger because the initial call causes a re render, and the patched element has its handler updated to use the stub method that you added to the instance.