Colle system, king's Indian attack, Reti double fianchetto...There's also the Jobava-Prie attack which is fairly new. I don't know how much theory has changed but there isn't much and new ideas are still being discovered.
An opening for white sought for (like Caro-Kann or French)

The Stonewall Attack. With such a blocked center, there is plenty of time to develop. Using the Stonewall Defense should also fit into you defense repertoire, and you can play ...e6 against everything.

I'd say almost any opening has lots of theory. But it doesn't really matter since none of us are GMs. Just don't play the Stonewall. It's a skill-less opening and people will laugh at you for playing it (I'll be one of them). Just play something like the Nimzo-Larsen, Hikaru plays it all the time in blitz, so you can always just watch one of the best crush people with it.

King's gambit has no theory. I know no theory, only a couple of ideas, and I have had a lot of success (and fun) with it.

I'd say almost any opening has lots of theory. But it doesn't really matter since none of us are GMs. Just don't play the Stonewall. It's a skill-less opening and people will laugh at you for playing it (I'll be one of them). Just play something like the Nimzo-Larsen, Hikaru plays it all the time in blitz, so you can always just watch one of the best crush people with it.
Hikaru most of the times transposes to hippo.
Some would say that hippo is even more skill-less than Stonewall.
That is not true. He only plays that sometimes as black, as white he always transposes into a d4 pawn opening. The OP is asking for an opening as white.
I noticed Kramnik has played a lot of Colle/Torre-Colle setups as White this year. Maybe because he doesn't want to spend a lot of time on theory and prep and the setup gives him a generally solid position.

Find a player whose repertoire as Black you like and emulate their choice as White.
Many of the great heroes of the French tended to play the Queen's Gambit as White.

Try the London System - it's basically the Slav or Caro-Kann in reverse with a move in hand.
I, a lifelong 1 e4 player, have recently tried out a couple of 1 d4 2 Nf6 3 Bf4 London System games because I play the Slav or Caro-Kann most often as Black. You've much of the same plan as the Caro-Kann or Slav: get-the-QB-out to the f or g file before moving your e-P one square and blocking it in. You've got the same QN to d2 (d7 in Caro-Kann/Slav) where it supports f3, e4, c4, and b3.
While you can set up the c3, d4, e3 pawn structure like the Caro-Kann's c6, d5, e6, you can often get the c-P to c4 in one move.
The Caro-Kann often gets the QB out at the expense of requiring two moves to get your P to c5, compared to the French where you move to c5 in one move but your QB is blocked in.
In the London System, since you are playing White and have a move in hand, you can often get c4 in one move.

<--- King's Indian Attack is a strong set-up.
Not much to memorize. Join our club and we'll show you how.
https://www.chess.com/club/kia-inc

There's also the Jobava-Prie attack which is fairly new.
I had to look this one up, since I never heard of it before!
http://www.chesspublishing.com/content/8/nov16.htm

Easy, just try to play what u like as black...
1.c3 or 1.e3.
Depends also what u play with black on 1.d4, since black is not forced to go for reverse Caro or French, and can go for 1.c3/e3 d5 for instance...
Jobava attack is indeed a good way to be lazy and it can be very effective.

"An opening for white sought for (like Caro-Kann or French)"
I think Fatman12 meant Fried Liver au gratin.

There's also the Jobava-Prie attack which is fairly new.
I had to look this one up, since I never heard of it before!
http://www.chesspublishing.com/content/8/nov16.htm
It's interesting in this day and age because unless Black is following games he's pretty much on his own.

Tracking off.
Many of these comments are just throwing out names of openings with no reference to the Caro-Kann or French.

Tracking off.
Many of these comments are just throwing out names of openings with no reference to the Caro-Kann or French.
Colle, London, Torre /thread I guess?

I think the initial question is a bit obtuse. The Caro-Kann and French are black defenses. They both respond to 1.e4; so how can a white opening be like them? Do you mean the reverse of those defenses? I think the initial question is not very clear.
So I don't think it's unusual for people to be throwing out all kinds of openings.
My own suggestion of the KIA was prompted by the fact that there are specific variations which deal with the French and the Caro-Kann. As well as a specific variation vs Sicilian and others which usually amount to a tweak in move order.
I am looking for a reliable opening for white which would not require much theory to learn (I don't want to spend much time on this) and would be based mainly on ideas and plans to understand (rather than on lines to learn by heart). In general I'd like it to be tactically safe (I don't want any fireworks too early) and to let me slowly and cautiosly develop my pieces, protect the king and, then, slowly start my attack.
As Black I find it optimal to play Caro-Kann or French (even though I try to play both of them as a safe basis for an attack!).
What would you recommend for White???