Can you outgolf Bill Gates?
Python 2 (PyPy), 238 235 222 bytes

a=input();n=len(a);r=range(n);a=zip(a,r);a=map(sorted(a).index,a)+[n]
def s(u,m):
 if m&lt;1:return[0]
 for k in r:
  v=u[k::-1]+u[k+1:]
  if sum(1&lt;abs(v[i]-v[i+1])for i in r)&lt;m:
   p=s(v,m-1)
   if p:return[k]+p
print s(a,5*n/3)

*(2 spaces = tab)
Try it online!
Saved 13 bytes borrowing a method for ranking a list.
DFS with a simple heuristic that checks whether a flip separates a pair of &quot;pancakes&quot; that would be adjacent when sorted. Sorts it in ascending order. Output is 0-indexed from the left where 0 flips the first 2 and so on. The number of moves used is (5/3)*n+1 &lt; 5/3*(n+1) where (18/11)*n &lt; (5/3)*n+1 &lt; 5/3*(n+1) and (18/11)*n is a tighter upper bound found in 2009.