Is it Islamically permissible to deal with an engineer by giving him a percentage of the construction cost as profit, which might take many long years, knowing that the agreement includes the type and cost of materials in advance, or is it considered a deception and an unknown sale, based on the fatwa of one of the shaykhs?

If the engineer constructs the building using materials he provides, the contract is one of "Istisna'a" (manufacturing contract), and it is permissible according to the Hanafis if its conditions are met. These conditions are: clarifying the type, kind, quantity, and desired specifications of the manufactured item, specifying the deadline, and it is permissible to defer payment. The Fiqh Academy has approved its permissibility.

If the engineer only undertakes the work, and the client provides the materials, the contract is one of "Ijarah" (lease/hire), and the wage must be known and specified. If the project cost is known, then there is no harm in the wage being a fixed percentage of it; otherwise, it is not permissible.