The Beirut dessert described in the question is daoukieh, also spelled daoukiyeh or daoukiyyeh. It is a semolina-and-pistachio sweet named after its inventor and traditionally served with ashta, the thick cream used in many Levantine desserts.
Daoukieh is associated with Al Daouk Sweets in Beirut. Its characteristic structure combines pistachio paste with a creamy filling, often including ashta and cashews. The dessert is particularly popular during Ramadan, when Beirut sweet shops prepare large quantities for customers buying treats after the evening iftar meal.
The dessert is sometimes compared with mafroukeh because both use pistachios, semolina and cream-like elements. They are not identical, however: daoukieh is recognized for its vivid pistachio layers surrounding the creamy center, while mafroukeh is generally prepared as a different pistachio-and-semolina confection.
Kellaj, barazek and mafroukeh are all Lebanese or Levantine sweets, but none matches the combination of Beirut origin and a name derived from its inventor as precisely as daoukieh.