This is already a known variant (modulo the constraint on the second move not relying on the first). Best way to play it is that the white's first turn is only one move, then black starts with the first double move - this is the 12* protocol (first implemented in a game called Connect6), and it practically eliminates any first move advantage
That sounds so fun. Honestly, I wonder how you could enforce that on more experienced chess players; is there a way to make a second move "forgetting" what you did first?
"Logically depend" has an easy interpretation: the second move must be also be valid if performed before the first move. [You could do a harder variant where both moves must be legal simultaneously.]
normal rules, except each side makes two turns in a row.
Constraint: the second move cannot logically depend on the first move in any way.
We've played a few dozen games this way. Really opens up some interesting possibilities.