Why doesn't Concorde stall when passing Mach 1, where a normal airliner does?
The phenomena you are referring to is better known as "tuck".  This is caused by buildup of a supersonic shock wave on the upper surface of an airfoil when a plane enters the transsonic Mach region of flight (approaching Mach 1).  This leads to a disruption of airflow behind the shockwave, resulting in a pitching down even without excessive AOA.  Concorde actually did this too, and trimmed for it with the help of fuel transfer pumps.  Discovery of this effect in the 1940s also lead to the creation of the "all flying" horizontal stabilizer. 

The designers gave Concorde a very long, pitch stable design, along with a delta wing.
That, and 4 powerful Rolls Royce/Snecma 593 Olympus engines, easily pushed it past transsonic, cruising along at Mach 2 up to 60,000 feet.

"Normal" airliners simply are not designed for supersonic flight.