Short answers, pulled from the story.
Thomas Cromwell established the Court of Augmentations in 1536 during the reign of King Henry VIII. This new body emerged to manage lands and finances seized from dissolved monasteries across England.
By 1540 all religious houses had gone with the last to fall being Waltham Abbey in Essex. Their lands properties and incomes went directly to the Crown after this final dissolution event.
The court operated to gain better control over land and finances formerly held by the Roman Catholic Church in the Kingdom of England. It administered estates of the crown while the term Augmentations referred to lands seized by Henry VIII when he dissolved the monasteries.
Richard Duke served as Clerk from 1536 until 1554 while Matthew Colthurst worked as Auditor between 1539 and 1552. Appointments were made by the Crown using letters patent under the great seal unless otherwise indicated these positions lasted for life.
In 1547 the Court of Augmentations merged with Court of General Surveyors which had been established in 1542 to administer crown lands. The court was incorporated into the Exchequer in 1554 as the Augmentation Office marking the end of independent operation after nearly two decades managing monastic spoils.