Nothing is as huge or playable to me as my 58 lester. That said, with budget in mind, an Epiphone Les Paul or Les Paul Special made in Japan at the Fujigen plant is a great place to start looking. Of the Tokai, Edwards, Burny and other well-known Japanese runs, I've found the Fujigen Epiphones with Gibson-shaped headstocks have the chunkier necks. ymmv
The Gibson faded SG also has a nice big neck, but know you're looking for a lester.