As pot prohibitions tumble state by state, companies are in a race across America for licenses and locations. The best funded of these cannabis contenders, Curaleaf Holdings, is a creation of Russian billionaires.

Curaleaf’s board chairman Boris Jordan, 52, played central roles in Russia’s messy return to capitalism in the 1990s. He helped arrange its first privatization, launched its first investment bank, and dove into the scrum to snatch up Russian assets—with mixed results for some backers like George Soros. Jordan’s bid...