here is a shooting method solution right out of the docs:

sol = First[
  NDSolve[{x''[t] + Sin[x[t]] == 0 , x[0] == x[10] == 0}, x, t, 
   Method -> {"Shooting", 
     "StartingInitialConditions" -> { x'[0] == 1.666 }}]]
Plot[Evaluate[x[t] /. sol], {t, 0, 10}]

The shooting method is of course iterative, so how to monitor progress (initial condidion vs end condition )? I came up with this sort of hack approach and I wonder if there is a better way:

define a function that always returns zero, but saves what we want as a side effect:

zero[t_?NumericQ, x_, xp_] := 
     (If[t == 0, xp0 = xp, If[t == 10, Sow[{xp0,x}]]]; 0)

now add as a term in the equation (no effect on solution since its always 0):

r = Reap[NDSolve[{x''[t] + Sin[x[t]] == zero[t, x[t], x'[t]] , 
      x[0] == x[10] == 0}, x, t, 
        Method -> {"Shooting",
         "StartingInitialConditions" -> { x'[0] == 1.666}}]][[2, 1]];

ListPlot[r, Frame -> True, FrameLabel -> {"x'[0]", "x[10]"}]

is there a better way? I tried working with WhenEvent with no luck..