The challenges facing the creation of brazed joints of 12Kh18N10T (AISI 321) austenitic stainless steel with high unbrazing temperatures for extreme working conditions in aerospace appliances are considered in this study. An amorphous-nanocrystalline nickel–based foil, Ni-7Cr-4.5Si-3.5Fe-2.6B, wt.%, is used for brazing the steel. Experiments on brazing regimes with various temperatures (1070–1160 °С) and times of exposition (15–80 min) are carried out. The formation of Ni-based solid solutions with different Cr, Fe, Mn, Si, and Ti contents in the brazed seam is detected for all brazing regimes. Using Thermo-Calc software, the liquidus temperature located in a range between 1268 and 1388 °С is calculated. To verify the calculated values, the unbrazing temperature is experimentally determined for specimens, with a composition of the solid solutions, formed in the center of the brazed seam during brazing at 1160 °С/15 min, 1070 °С/15, and 1160 °С/80 min. The experimental results deviate from values computed by Thermo-Calc by no more than 54 °С. The experimental samples obtained using the 1070 °C/40 min and 1100 °C/80 min regimes demonstrate high unbrazing temperatures equal to 1303 °C and > 1330 °C, respectively, which corresponds to the calculations with an accuracy of 2.5%. Taking into account the combination of properties (strength and unbrazing temperature), brazing regimes of heating to 1070–1100 °C and an exposure time for 15–40 min can be recommended for the production of high-strength joints with high unbrazing temperatures.