Applying Function with SlotSequence and Lists

This gets the desired result:

(MyFunction @@ #1) @@ #2 & @@ {{1, 2, 3}, {4, 5, 6}}
(*  MyFunction[1, 2, 3][4, 5, 6]  *)

User Michel E2 has already given a good answer. An alternative is to use (hat tip to Kuba for their suggestion):

Fold[Apply, MyFunction, {{1, 2, 3}, {4, 5, 6}}]

which evaluates to the same expression.

I'm not sure what exactly your goal is, but note that the code above works for lists of arbitrary length:

Fold[Apply, MyFunction, {{1, 2, 3}, {4, 5, 6}, {7, 8}, {9}}]
(* MyFunction[1, 2, 3][4, 5, 6][7, 8][9] *)

For fun: in the case of two lists, you can also use pattern replacements:

{{1, 2, 3}, {4, 5, 6}} /. {{a__}, {b__}} -> MyFunction[a][b]

which yields, once again, the same output.