When the delegates at the Philadelphia Convention designed the Constitution they debated the structure of government and how to best implement a proper separation of power and system of checks and balances. When discussing Article II there were concerns about the office of the Presidency and its relationship with the executive council. By this point most states had an executive council also known as the privy council or the governor's council. There were various ways to structure the council. You can have the executive council be appointed entirely by the legislature without the chief executive's approval, the council could be made up of members of the legislature, the chief executive could appoint advisors by themselves, the chief executive could nominate advisors who would need approval from the legislature, etc. The delegates decided on the last option but the relationship was still not clearly enumerated and the first few presidential administrations consisted of debates between the two branches over the Cabinet. The Presidents and the Senate wanted as much control and oversight as possible.
I do not believe that the current structure of the President's Cabinet is well designed and so here is my proposed amendment.
Each department will be headed by two Consuls, one styled as a Secretary and the other a Chief. The Secretary is nominated by the Speaker of the House and approved by the Senate. They can only be removed by impeachment/conviction or by a joint resolution. The President cannot reject or remove a Secretary but could appeal to either house with a removal request. The Secretary will advise the President and must be in the same room whenever the President and the department's Chief are meeting. The Secretary can be called by the Congress to give a report on the activities of the department and Congress can request or pass a bill to change specific department's policies.
The second Consul is the Chief of the department. They will run the operations and management of the department and subsidiary agencies in a more executive manner. They will enforce the policies determined by the President and Secretary while also gathering data about its department's operations to be relayed to the President and Secretary. The President will have full discretion in appointing the Chiefs and do not require Congress' approval. The President may remove, replace, or add Chiefs at will. Congress cannot. Congress can impeach a Chief via House and conviction via Senate for committing high crimes, treason, or indecent misdemeanors unbecoming of an advisor to the President.