I do understand what you are saying and it's not what Hinduism, at least my sect, teaches.
From Dancing with Shiva:
"Rishis proclaim that we are not our body, mind or emotions. We are divine souls on a wondrous journey. We came from God, live in God and are evolving into oneness with God. We are, in truth, the Truth we seek.
We are immortal souls living and growing in the great school of earthly experience in which we have lived many lives. Vedic Rishis have given us courage by uttering the simple truth, “God is the Life of our life.” A great sage carried it further by saying there is one thing God cannot do: God cannot separate Himself from us. This is because God is our life. God is the life in the birds. God is the life in the fish. God is the life in the animals. Becoming aware of this Life energy in all that lives is becoming aware of God’s loving presence within us. We are the undying consciousness and energy flowing through all things.
Deep inside we are perfect this very moment, and we have only to discover and live up to this perfection to be whole. Our energy and God’s energy are the same, ever coming out of the void. We are all beautiful children of God. Each day we should try to see the life energy in trees, birds, animals and people. When we do, we are seeing God Shiva in action. The Vedas affirm, “He who knows God as the Life of life, the Eye of the eye, the Ear of the ear, the Mind of the mind—he indeed compre hends fully the Cause of all causes.”
It's a Panentheistic way of looking at the world. God is life, He is the life force in everything. We come from God and are God. It's just that we have to realize it. It's not God being cruel; it's Him showing us that we are not this body. We are so much more than we think we are. This is especially important to remember in Kali Yuga, the age of Dharma decline, that we are in right now. People think they are different or seperate from God, but Maya (illusion) is what causes that. When we break the illusion, we see that we are God.
You keep asking if we are God's lost parts; no. Like it says above, the one thing God cannot do is seperate Himself from us. We are still one with Him, we just need to remember. One day, everyone will remember.