This article is one in a series of studies on Elohim, one of the Hebrew names for God - you are currently on the topic that is in bold:
El Shaddai is found seven times in the Scriptures (Gn 17:1; 28:3; 35:11; 43:14; 48:3; Ex 6:3; Ez 10:5), while Shaddai appears more frequently (in Job 30 times, in Psalms 19:1 and 68:14, once in Isaiah (13:6), Ezekiel (1:24), Joel (1:15), and Ruth (1:21)).
El Shaddai most likely means "Sovereign one" or "All Powerful", rather than "breasted one" as William Branham taught.
In the passages referred to above, the combined ideas of God as the all-powerful, all-sufficient, transcendent, sovereign ruler and disposer are present. This meaning is generally accepted. The most preferred explanation is that Shaddai is derived from the verb shadad (“to overpower, to deal violently, or to devastate”). A clear connection between shadad and Shaddai is said to be found in Isaiah 13:6 and Joel 1:15. God as El Shaddai is presented as the all-powerful One, totally self-sufficient, absolute ruler, and the One who can and does make final disposition.
The Septuagint (LXX) has adopted this meaning; it translates El Shaddai as Pantokrator, the “All Ruler or Sovereign One.”
G. Vos points out that El and Elohim emphasize God’s relation to nature, and El Shaddai points to God’s overpowering nature in the service of his grace and compelling her to further her designs. L. Berkhof (Systematic Theology) points out that whereas God as El Shaddai is indeed presented as the all-powerful One who overpowers nature, the name, where it occurs in the Bible, does not present God as an object of fear or terror, but rather as a source of blessing and comfort.
Some suggest that Shaddai is a composite term of sha (“the one who”) and dai (“is sufficient”). Some later Greek versions of the Old Testament have adopted this meaning. This view is not as well accepted as the view above.
Others believe that Shaddai may be an archaic form for shade = “mountain,” and that El Shadaii may mean “dweller on the mountain.” The “inhabitant of the mountain” would easily become “the mighty one” or “the almighty one,” in consequence of the fixedness of the mountain and the impregnable character of the sanctuary.
The name Ṣuri-shaddai (“my rock is Shaddai”), as well as the later use of ṣuri as an epithet of Yahwe in Ps. 18:2, 31:3, and 2 Sam. 22:2, would add strength to this. Yahwe originally appeared in a mountain — Horeb — so that the two are kindred.
Least accepted meanings
Some have begun with shad as the first concept to be considered; its meaning is “breast, pap, or teat,” and it is considered by them as a metaphor of God who nourishes, supplies, and satisfies. However, the root of shad (shadah) in Semitic usage, is to moisten; this meaning is not appropriate in the context in which El Shaddai appears; nor is shed (demon), which some scholars have sought to use because it appears in Deuteronomy 32:17 and Psalm 106:37 speaking of Israel’s idolatry. In addition to the fact that shed is spelled differently, the connection between the concept of demon and God as all-powerful is difficult to establish.
William Branham took his reference to the breasted god from the Scofield Bible. Scofield was not a Hebrew scholar. It is interesting to note that the recent Scofield revision (1967) recognizes the error of the prior versions and focuses on the meaning as “all sufficient” and to the usual translation of EL SHADDAI as “God Almighty.” 
William Branham's use of the Scofield Bible
The Emphatic Diaglott is a Greek Interlinear version of the New Testament published by the Watchtower Society, so William Branham's suggestion that it contains a reference to Genesis 17 is specious. Additionally, William Branham admits that he did not use the Thompson Chain Reference Bible, so his reference to it is hypothetical and based on assumption. The Bible that he admits to using almost exclusively is the Scofield Bible. The "Thompson Chain" is only referred to by Branham on 4 occasions and always in the same paragraph as a reference to the Scofield Bible.
Interestingly, William Branham plagiarized Clarence Larkin's works on a significant basis but only mentions him 3 times in all of his sermons and never by attribution of an idea that he took from Larkin. Scofield, on the other hand, is mentioned over 40 times.