[verb ます-stem] + かけ + の + [noun] [verb ます-stem] + かけ + だ [verb ます-stem] + かける
かけ after a ます-stem describes something started and then left unfinished: a half-eaten cake, a half-read book. It behaves like a noun, so it takes の before another noun and だ or です at the end of a sentence.
The verb is in the ます-stem, never the dictionary form.
What かけ means depends on the verb. With an action that takes time – eating, reading, writing, making – it is "partway through". With a sudden change – breaking, dying, going out – it is "nearly": the change has started and hasn't happened yet.
かけ is the noun form of the verb suffix かける, which takes the same ます-stem and conjugates as a る-verb: かけた, かけて, かけている. かけ describes the thing left in that state; かける says what someone did or what nearly happened.
With ている, a change is under way now and hasn't gone through yet.
In はなしかける話しかける "speak to someone", よびかける呼びかける "call out to" and わらいかける笑いかける "smile at", かける points the action at a person, and nothing is half done. These are verbs in their own right, and the person takes に.