How to begin Buddhist-Christian dialog? Get the Buddhists to begin a lecture series on Mindfulness. Then offer fellowship workshops (with coffee and desserts, of course) in which Christians write short tracts about the Mindfulness displayed by Jesus; and of course, compose some catchy hymns in which everyone sits in meditation-pose singing with arms waving overhead.
Seriously, a theological middle ground (sounds like Buddha's "middle way"!) can be sought with Christian theologian Paul Tillich's God as the "ground of being," which is the very description sometimes given of Buddha; and the writings of Christian monk-scholar Thomas Merton ( lived at Gethsemane monastery in Kentucky), who was influenced by Zen.
Rather chilly Buddhism needs more of the personalization of the deity through the life of Jesus offered by Christian thought; and Christian thought needs to move beyond the anthropomorphic notion that God is like a rather removed, unknown, egotistical father residing somewhere else playing chess with the world, and whose moves can be influenced by prayer. There are lots of examples of Christian mysticism through more than a millennium that can be used as textual examples of that which the sacred writings point.
In contrast, every time I go to a Christian service (last attendance was Easter Sunday) I find it terribly distracting with words, words, words. So, Christians could learn that the logocentricism of Christian praxis (scripture, sermons, hymns, Christian witness, etc.) does not get anywhere near the great Reality we all live within. Church services are more entertainment and stimulus (for the mind and emotions) than "worship" service. That takes stopping the words. It takes Mindfulness. Perhaps the Mindfulness of Jesus.