Medicinal and aromatic plants continue to attract attention as rich sources of natural bioactive compounds with potential health benefits. Among them, Greek mountain tea (Sideritis scardica) is widely recognized for its high content of phytochemicals, which have been associated with various biological activities. In this study, Natural Deep Eutectic Solvents (NADESs) were investigated as a sustainable and efficient alternative to conventional solvents for the extraction of such compounds, aiming to the development of a more efficient extraction process. Six task-specific designed NADESs were prepared and evaluated for their extraction efficiency, based on the Total Phenolic Content (TPC) and Total Flavonoid Content (TFC) of the extract. The most promising NADES, comprising betaine and 1,3-propandeiol, was selected for process optimization using a Box–Behnken design and key extraction parameters were systematically examined to maximize TPC and TFC. The extract obtained under the proposed conditions (S/L = 20 mg/g, 240 min, 40% water as cosolvent) exhibited a TPC and TFC value of 49.2 mg_GAE/g and 45.9 mg_CAE/g, respectively, approximately two times higher than the values of a hydroethanolic extract, obtained under the same conditions (TPC = 26.6 mg_GAE/g, TF = 19.9 mg_CAE/g). The optimum extract was further analyzed using HPLC-DAD to determine its phytochemical profile and was compared with the conventional hydroethanolic extract, revealing the role of the selected media on the extracted compounds.