With Mockito, how do I verify my lambda expression was called?

You can verify that your method was called with any lambda expression was called like this:

    verify(someClass).myMethodThatExpectsALambda(any())

    private fun <T> any(): T {
        Mockito.any<T>()
        return null as T
    }

Some of the other answers offer alternatives to doing exactly what I want here, but this is doable by Spying the Consumer class itself and having the spy call the method you really want to execute. A helper method for wrapping the lambda to create the spy helps here:

/** Get a spied version of the given Consumer. */
private Consumer<Item> itemHandlerSpy(Consumer<Item> itemHandler) {
  // Create a spy of the Consumer functional interface itself.
  @SuppressWarnings("unchecked")
  Consumer<Item> spy = (Consumer<Item>) Mockito.spy(Consumer.class);
  // Tell the spy to run the given consumer when the Consumer is given something to consume. 
  Mockito.doAnswer(it -> {
    // Get the first (and only) argument passed to the Consumer.
    Item item = (Item) it.getArguments()[0];
    // Pass it to the real consumer so it gets processed.
    itemHandler.accept(item);
    return null;
  }).when(spy).accept(Mockito.any(Item.class));
  return spy;
}

And then the test method becomes very straightforward:

Consumer<Item> itemHandler = itemHandlerSpy(Item::foo);
instance.conditionalRun(itemHandler);
// This verifies conditionalRun called the Consumer exactly once.
Mockito.verify(itemHandler).accept(Mockito.any(Item.class));