The common theme of this collection is that the poet's thirst for meaning and God in this world leads him to remember the divinized imagery of his parents in his childhood memories of them. For example, in "The Cleaver," the speaker's father gossips like his own mother, that shows a direct parallel to this theme, but even more so later in the poem, when the boy eats shellfish, he notices that it was he was connected with the food in a deep, intimate way. So in this poem, the boy is shown the secret meaning of life by his parents, that life survives by the death of lower life forms that we call animals.
This seems perfectly obvious, but it is the function of poetry to allow people to see the magic quality of ordinary things, so look closer at the imagery. The title of "The Cleaver," is a reference to a rending quality, something that separates one part from the rest of a thing, namely the butcher who cuts apart animals to make them into food. But more importantly, the real cleaving has already been done. By identifying himself to the animals he is eating, he participates in a sacrament akin to communion. He communes with the living power that sustains all powerful life. This is a religious epiphany, and where is it located in the poet's mind? Near the imagery associated with his parents in his memory.
This means that to realize one's own mortality depends on the imagery of parent mortality, and we get that in "Early in the Morning." What happened early in the morning? The poet sees his mom's hair falling out of her head like drapes. His mother is dying. He realizes it early in the morning. So in his mind, death is a thing to be thought about in the quiet privacy of a morning with mom and dad. It's intimate and familial, and it's deeply personal. Death doesn't look like a demon or a carcass to the poet. The poet sees his own mother when he thinks of death. There is the center of the poignancy of the collection.