The board of directors performs the dual role of monitoring and advising the firm's management, and at times makes certain key decisions itself. We study the optimal board composition (of monitoring and advisory "types") within a cheap-talk framework where the CEO and the board each may have private information about an impending investment decision, and their incentives are imperfectly aligned. When shareholders choose both the board composition and the allocation of decision rights between CEO and board, a non-monotonic relationship between CEO bias and board composition emerges. Counter to conventional wisdom, we show that powerful CEOs who nominate board members themselves may in fact prefer a greater degree of monitoring intensity on the board than the shareholders do. As a result, regulatory interventions (such as the Sarbanes-Oxley Act) that attempt to strengthen the compliance role of boards, will in fact be harmful in precisely those cases where agency problems are the most severe. Lastly, CEOs may be able to entrench themselves by choosing "complex" projects involving greater information advantage. In response to the threat of CEO entrenchment, shareholders may commit to a board composed of mostly advisory types.