I took a look at a dozen Israeli companies that do business in the West Bank settlements, and between my experience and that of other members of the questioner's group, here is an "answer" to this query.
As the questioner said, the Who Profits? project is the best starting point for researching connections between businesses and the Israeli occupation industry. The data on this site is compiled by The Coalition of Women for Peace, "a leading Israeli feminist peace organization, dedicated to ending the Israeli occupation of the West Bank, Gaza and the Golan Heights and reaching a just peace in Israel/ Palestine" (from whoprofits.org/About.php).
Who Profits will give you names of CEOs, owners, parent companies, and subsidiaries. You can then search-engine them (as well as the names of the companies, of course) to see what comes up on the open Web. I added the terms "new york" and "brooklyn," plus terms related to the business in question, as needed to help channel the results into something relevant. For example, "'eyal peled' 'new york' engineering" (Eyal Peled is one of the owners of Peled-Klein Civil Engineering).
I also tried a number of subscription databases. Thomson Research and Bureau van Dijk Electronic Publishing's Osiris had profiles on the one publicly traded (on the Tel Aviv Stock Exchange) company in the batch I worked on, and that information includes members of the board of directors and some institutional shareholders.