There are really two standard introductory works on the subject, the old Halil İnalcık, The Ottoman Empire: The Classical Age, 1300-1600, translated by Norman Itzkowitz and Colin Imber (London: Weidenfeld & Nicolson, 1973) and the book meant to serve as its replacement, Colin Imber, The Ottoman Empire, 1300-1650: The Structure of Power, 2nd ed. (Hampshire: Palgrave Macmilllan, 2009). Despite its age, İnalcık is still very strong on Ottoman institutions during this period (not so much other aspects of the book), which is why I still find it worth recommending. İnalcık's term "Classical Age" for the period from Mehmed II to Süleyman I has had a long shelf life. To move on to more narrow studies, Gülru Necipoğlu's Architecture, Ceremonial, and Power: The Topkapı Palace in the Fifteenth and Sixteenth Centuries (Cambridge: MIT Press, 1991) explores the history of the construction and function of the imperial palace, essential reading insofar as the government of the empire can be equated with that of the imperial household in these centuries. Likewise, Leslie Peirce's The Imperial Harem: Women and Sovereignty in the Ottoman Empire (Oxford: Oxford University Press, 1993) is another standard in the field for demystifying the role of women in the Ottoman imperial household, and the politics of dynastic reproduction. Linda Darling's Revenue-Raising & Legitimacy: Tax Collection & Finance Administration in the Ottoman Empire 1560-1660 (Leiden: E. J. Brill, 1996), while focused primarily on a later period, begins with an account of the functioning of the Ottoman bureaucracy in the period you're interested in. For provincial administration, Metin Kunt's The Sultan’s Servants: The Transformation of Ottoman Provincial Government, 1550-1650 (New York: Columbia University Press, 1983) is essential. Like Darling, he focuses on a later period, but begins with an account of the nature of provincial government in the "Classical Age." The Arab provinces are covered by Jane Hathaway in The Arab Lands under Ottoman Rule, 1516-1800, with contributions by Karl K. Barbir (Pearson Education Limited, 2008), and she offers additional suggested readings as well. On law, social institutions, and a myriad of other topics, the essays in Christine Woodhead ed. The Ottoman World (New York: Routledge, 2011) are worth a look. More recent studies have given us a fuller picture of legal institutions in the sixteenth century, see Guy Burak, The Second Formation of Islamic Law: The Ḥanafī School in the Early Modern Ottoman Empire (Cambridge: Cambridge University Press, 2015) and Abdurrahman Atçıl, Scholars and Sultans in the Early Modern Ottoman Empire (Cambridge: Cambridge University Press, 2017).