Mild Cognitive Impairment (MCI) and Alzheimer's diseases (AD) are two common neurodegenerative disorders which belong to the dementia family mostly found in elders. There is evidence that MCI may lead to Alzheimer's disease. Since there is no treatment for AD after it has been diagnosed, it is a significant public health problem in the twenty-first century. Existing classical machine learning methods fail to detect AD and MCI more efficiently and accurately because of their shallow and limited architecture. Electroencephalography (EEG) is emerging as a portable, non-invasive, and cheap diagnostic tool to analyze MCI and AD, whereas other diagnostic tools like computed tomography, positron emission tomography, mini-mental state examination, and magnetic resonance imaging are expensive and time-consuming. To address these obstacles, a deep residual Alzheimer's disease and MCI detection network (DRAM-Net) based framework has been introduced to detect MCI and AD using EEG data. This multi-class study contains EEG data collection, preprocessing (down-sampling, de-noising and temporal segmentation), DRAM-Net architecture to classify AD, MCI and normal subjects and experiment evaluation stages. Our proposed DRAM-Net framework has obtained 96.26% overall multiclass accuracy, outperforming existing multi-class studies, and also claimed accuracy of 96.66% for the normal class, 98.06% for the MCI class, and 97.79% for the AD class. This study will create a new pathway for future neuro-disease researchers and technology experts.