Verified answer

You definitely have warped brake rotors, get the pads replaced along with the rotors (possibly also the calipers). The rotors are the circular discs the pads get pushed against to create the friction to stop, stomping on the brakes will cause them to warp, especially if you have been driving for a long period of time, they will be hot and that much more likely to warp. (Calipers are the moving parts that press the pads against the rotors, sometimes they start sticking and will keep one or both sides of the brake pads against the rotor as you start off, calipers need replacing if you have to really push your car when starting up from a stop, otherwise they should be fine)

But if your dash is vibrating an unnatural amount due to this, it could be that a previous owner, or a shop removed the dash at some point and either broke or lost some of the clips holding the dash in place.

The problem is.

The break rotors are warped.

Go to a nearby workshop