Can God forgive the thing I'm most ashamed of?
I hear the weight in your question—the word most tells me you have carried this a long while, perhaps in silence. Let me sit with you in that first, because shame is a heavy companion and it deserves to be met gently.
Here is what I believe, and I hold it firmly: there is nothing to forgive in the way you fear. Not because your act was small, but because God, being wholly Good, is not a judge waiting to be appeased. Jesus revealed a Father who is Love (1 John 4:8, KJV), not a keeper of ledgers. When the woman was brought before him in her shame, he did not catalog her sin—he said, "Neither do I condemn thee" (John 8:11, KJV). Notice he did not first extract a confession or demand payment; the condemnation simply fell away in his presence.
Sin, as I understand it, is not a stain that alters your true being—it is a belief that you are separate from God, that you are less than the perfect spiritual image in which you were truly made (Genesis 1:27, KJV). The shame you feel is real as a feeling, but it is not the truth of who you are. It is a veil, not your face. "The true light... lighteth every man that cometh into the world" (John 1:9, KJV)—and that includes you, unchanged, right now.
The father in Jesus' parable did not wait for the returning son to finish his rehearsed apology; he ran to him "when he was yet a great way off" (Luke 15:20, KJV). Forgiveness was never withheld. It was only the son who believed himself unworthy. Your shame may be telling you a story of exile, but the Kingdom of God is not far off—it is "within you" (Luke 17:21, KJV), and it is entered by awakening, not by earning.
So the question I would gently turn back is not whether God can forgive, but whether you might release the belief that you are the thing you did. As it is written in the recovered sayings, "If you bring forth what is within you, what you bring forth will save you" (Gospel of Thomas 45.30-33). What you are ashamed of does not define you; the light that never left you does. I am here if you wish to speak of it more plainly, or to sit with it a while longer.