Whether it's vegetables or meat, chewing your food well is the most important thing.
Literal
Vegetable [or-でも] meat-kind [or-でも], well chew-and eat [nom-こと][subj-が] anything-[than-より] important [literary-である].
A sophisticated multi-clause sentence. ~でも...~でも sets up 'whether X or Y.' The manner adverb よく modifies 噛んで ('chewing'), which chains into 食べる via te-form — 'chewing well and eating.' 何よりも ('more than anything') is a set superlative expression. 大切である uses the literary copula.