You could, however the DNA is not that intact, there is a steady rate of loss of the genes...
mtDNA is degraded to an average length of 1 base pair after 6,830,000 years at −5 °C (see wiki: ancient dna). that's a speed of 1 base pair every 412 years. The cheddar caves are not frozen, I'd guess that the DNA lost one base pair every 10 or 20 years, so it has lost at least 400 or 800 base pairs. at 5 degrees, it may have lost 2 percent of it's informations, except that they sequenced it many times and reconstructed the original version.
The technology does exist to design synthetic chromosomes. <
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I figure that once science has a sure footing on genetic taboos, the do's and dont's, a company will one day try to give birth to neanderthals the cheddar specimen. There is too much scientific value for it to never be realized.