Well, let's look into this mathematically:
https://www.livestrong.com/article/532479-nutritional-value-in-gingerbread-men/
I don't think we care about calories, but I DO think we care about mass!
"Calories in gingerbread men cookies vary depending on the size of the cookie. A large 3-ounce cookie contains 340 calories, while a smaller 1.4-ounce cookie contains 160 calories."
1.4 ounces("Smaller")=39.6893324 grams 3 ounces("Large")=85.0485694 grams
In the original fable, the Gingerbread man was outrunning even horses. Surprisingly, our wiki lacks a page for horses (Despite having pages for dogs, cats, cows & wild boars. Seriously?! Although, pigs probably meant domestic/livestock pigs.) so let's go with what The Gingerbread Man's profile says:
Our page for the Gingerbread man cites
http://www.speedofanimals.com/animals/horse which says 54.7 mph, or, if you put it in metric, 88 km/h. Google converts the former to 24.453088 m/s, & the latter to 24.4444444 m/s.
So, we have a cookie of
somewhere between 39.689 & 85 grams moving at
24.44 to 24.453 m/s.
Fun. Time to make 3 ends, I guess. I'll ask Wolfram, since I'm lazy. One with minimums, one with averages, & one with high ends. (I'll use the horse speed range for speeds in these 3 ends.) I have no idea if The Gingerbread Man qualifies as a "smaller" or "large" Gingerbread cookie.
(Also, a
Reddit Respect Thread cites this:
https://greggs.co.uk/sweet-treats/gingerbread-ma )
Low End: 39.689 grams moving at 24.44 m/s = 11.85 joules
Mid End(Averages): 62.3445 grams moving at 24.4465 m/s = 18.63 joules
High End: 85 grams moving at 24.453 m/s = 25.41 joules
So in all 3 cases, solidly Below Average Human.
And in fact, if I recall the fable correctly, the Gingerbread Man was killed by being flipped into the air & snapped up by a fox.
There are measurements of fox bite forces documented out there, but I dunno how to convert newtons or bite force quotients to our system. If anyone knows how, please inform me.
So, let's look at our Wiki's only page for a species of fox:
https://vsbattles.fandom.com/wiki/Fennec_Fox 10-C, albeit, with no calculation.
Supposing the species of fox that killed The Gingerbread Man with a bite is comparable to a fox, that'd mean he can be one-shot, or come very close to it, with 10-C force.
Can a fox bite with greater than 10-C force? How many joules is it?
The reason this is relevant is because if a 10-C biting him is enough to split TGM in half, killing him, then something 11.85 to 25.41 joules (Roughly 25% to 50% of maximum 10-C force.) could almost certainly kill him, too.
And as it so happens, those are just the KE yields of him ramming himself. What I'm saying is there's a good chance Lil' Gingy ramming himself into Burns to attack would be suicide, as he's canonically died to comparable forces.
Mr. Burns has technically survived being thrown several meters back by a baseball hitting a baseball bat he was holding, & he's been shown to be able to hold & move a bat.
He might also be strong enough to take off his jacket & throw it on top of The Gingerbread Man, & "elbow drop" (Read: Fall on) him, too! Clothes covering could help immobilize him or blind him, or weigh him down so he can better be caught & snapped up.
Burns is also smarter, & apparently has better Lifting Strength.
I think the best case is Gingy rams into Burns, breaking himself, & the force knocks out or severely injures Mr. Burns.
Is killing yourself to knock your opponent out Inconclusive or a loss?
I'm voting Mr. Burns.