I think I get a t-shirt on this one.
You may. After recalling that transport-class aircraft can have more than two engines

, I went back and edited my original post. I designed for Boeing, which mostly means twins. After more careful thought I stepped back.
Further, I neglected to consider regional jets, which have impressive thrust and can probably sustain nominal climb rates on one engine.
All transport category multi-engines can climb on one engine at max certificated gross take off weight after the loss of an engine (unless there's an exception in the regs that I missed).
You have to read the regs carefully, especially where it discusses which takeoff stage the requirements apply to. FAR part 25 requires a gear-down, full thrust climb gradient that is merely "positive."
How positive in practice depends on the design and loadout. My recollection from the medium-sized Boeing airframes is first-stage OEI gradient disappointingly less than 1%. I know the new 787 improves on that significantly, and I'll defer to practical experience on the other airframes. Getting to a point where you can retract the gear is paramount. See below.
Before you rotation, you have options. After you lift off, you have options. Right at rotation is where FAR part 25 basically just says, "Do your best, at pilot's discretion."
Are they easy to control with all the thrust being generated from one side only, and a considerable distance away from the central axis?
Not a problem. Boeing twins have the engines mounted as inboard as possible so that off-axis thrust is well within the aerodynamic yaw control capability, even with one engine at takeoff thrust, the other at flight idle or inoperative, and airspeed at merely VLOF (i.e., where rudder effectiveness is diminished compared to cruise or climb flight).
Don't forget, anything with a turboprop and above, and especially swept-wing jets, have yaw dampers that will assist the crew in keeping the plane from yawning into the dead engine (and the resultant coupled-roll).
[makes the elongated "eeeehhhhh" noise one makes when wondering how true something really is]
In Boeing control systems the yaw damper inputs are limited to a fraction of available blowdown. This is to prevent automatic rudder overcontrol. Basically the yaw damper will provide immediate reaction to
uncommanded yaw, but a successful recovery will require pilot rudder commands to achieve an effective yaw correction rate.
Yaw-damping in modern FCCs is based around detecting and correcting cycles, such as in Dutch rolls, so as not to interfere with pilot sensitivity. As such, strict yaw-damping is often tied into the roll channel as well and may not activate immediately until a roll develops (i.e., the wing with the dead engine will drop). That said, yaw control in FCCs also includes yaw-stability controllers that provide yaw-only corrects, even if you don't classify it under yaw-damping for cyclical motion (as it was in the old days). And keep in mind that a big ol' vertical stabilizer is there to give you corrective yaw moments passively!
My takeoff brief is simple: "If we lose one and the gear's still, down, both throttles and mixtures go to idle-cutoff, we pitch for best glide, and land straight ahead, trying to avoid obstacles."
My first activity at Boeing was on braking systems. Back in college I designed, among other things, field-replaceable and -manufactured brake components for small defense airframes. When confronted with the awesome robustness of large airframe brakes, I asked for the rationale and was told that accelerate-stop distance (runway distance required to accelerate to VLOF, then brake to a full stop) was the dominating design factor since OEI accelerate-go performance (horizontal distance required to take off and climb to 50 feet, i.e., to clear any field clutter) was disappointing. The intended pilot response to sudden engine failure after rotation but prior to liftoff was a rejected takeoff, albeit incurring a (literally) flaming mass of brake assembly at the end. What operators might instead train their pilots to do based on operational experience is something I can't speak authoritatively about, so distribute T-shirts as appropriate.
Again, that's for twins and especially twins that aren't the 787. I helped design that airframe, and I'm immensely proud of its low-airspeed lift performance.
I can't heap enough praise on the L-1011. Lockheed did a wonderful job. But it does have three engines...
And for the 747 the response is: (mock English accent added for exaggerated nonchalance) "Oh dear, we seem to have lost an engine. Do make a note of it, Nigel." Okay, maybe not, but you get the idea. FAR part 25 for first-phrase climb on a 4-engine OEI is essentially nominal climb.