But how to describe it without language?
Also, doctrine is not totally irrelevant. I think it's part of "faith seeking understanding" - a conscious working through issues.
Consider the longing for G-d. A recurring theme for me is a sense of unconsolable loss and unresolved grief. This actually sounds a lot like Sufism:
An analogy for bodhichitta is the rawness of a broken heart. Sometimes this broken heart gives birth to anxiety and panic, sometimes to anger, resentment, and blame. But under the hardness of that armor there is the tenderness of genuine sadness. This is our link with all those who have ever loved. This genuine heart of sadness can teach us great compassion. It can humble us when we’re arrogant and soften us when we are unkind. It awakens us when we prefer to sleep and pierces through our indifference. This continual ache of the heart is a blessing that when accepted fully can be shared with all.
~Pema Chodron
My theistic solution to this is based on my understanding of the problem: we feel isolated/fragmented because we were separated from the Creator when we entered the world of finite forms. Being a finite creature means being fragmented. Our losses in this life remind us of the underlying and pervasive sense of loss and sense of being incomplete.