Lymphangioleiomyomatosis (LAM) is a rare lung disease that is caused by genetic mutations. It results in the uncontrolled growth and proliferation of an atypical smooth muscle cells in the lung. These cells invade airways, blood vessels, and lymph vessels, and limit the flow of air, blood, and lymph, respectively. The source of the cells is unknown, but available evidence indicates they arise from an extrapulmonary source. Their aberrant behavior is due to mutations in tuberous sclerosis genes that results in mTOR activation. Respiratory failure, lung collapse (pneumothorax), and pleural effusions (chylothorax) are hallmarks of the disease. This study will evaluate the safety and durability of the mTOR inhibitors sirolimus and everolimus, which are FDA approved medications for prevention of rejection of transplanted organs, in stabilizing or improving lung function in people in LAM.