That clause of the
U.S. Constitution reads as follows...
He shall have Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two thirds of the Senators present concur; and he shall nominate, and by and with the Advice and Consent of the Senate, shall appoint Ambassadors, other public Ministers and Consuls, Judges of the supreme Court, and all other Officers of the United States, whose Appointments are not herein otherwise provided for, and which shall be established by Law: but the Congress may by Law vest the Appointment of such inferior Officers, as they think proper, in the President alone, in the Courts of Law, or in the Heads of Departments.
... I’ve taken the liberty to highlight the relevant section which can render it Constitutional for a Special Counsel to be appointed by someone other than the president and without the consent of the Senate. That, however, would require the Congress to have passed such a law — which they did...
CFR › Title 28 › Chapter VI › Part 600: § 600.1 - Grounds for appointing a Special Counsel
The Attorney General, or in cases in which the Attorney General is recused, the Acting Attorney General, will appoint a Special Counsel when he or she determines that criminal investigation of a person or matter is warranted and -
(a) That investigation or prosecution of that person or matter by a United States Attorney's Office or litigating Division of the Department of Justice would present a conflict of interest for the Department or other extraordinary circumstances; and
(b) That under the circumstances, it would be in the public interest to appoint an outside Special Counsel to assume responsibility for the matter.
So it appears it was Constitutional for Rosenstein to appoint Mueller as Special Counsel as he did. The only argument against that would be if you could prove that Mueller is a principal officer since the Constitution only allows for the Congress to draft such laws as title 28 chapter VI § 600.1 if they apply to inferior officers.
Steven Calabresi of Northwestern University Pritzker School of Law tried to establish that argument but he failed miserably. Let’s see if you can do any better than a law professor....