We study some fundamental properties, such as the transience, the recurrence, the first passage times and the zero-set of a certain type of sawtooth Markov processes, called extremal shot noise processes. The sets of zeros of the latter are Mandelbrot's random cutout sets, i.e. the sets obtained after placing Poisson random covering intervals on the positive half-line. Based on this connection, we provide a new proof of Fitzsimmons-Fristedt-Shepp Theorem [FFS85] which characterizes the random cutout sets.