Green-Chile Burgers With Fried Eggs

I think this is the first of the F&W staff favorites that disappointed us a bit.  It was alright but not particularly memorable.  We did alter it a bit.  I only used a pound and a half of meat.  The burgers were plenty big with less meat.  The only blue cheese at the market was  danish style and probably stronger than what was recommended, but the burgers weren't overwhelmingly blue cheesy.  I don't care for brioche and we had some buns already.  And lastly, there weren't the right peppers at the market and a quick search on my phone suggested cubanelles as a substitute.  It was tasty but I can't see us making it again.  Maybe we have become a bit too picky, but we have plenty of recipes we love so... It was different and that is something:)
Green-Chile Burgers With Fried Eggs
2 1/4 lb ground chuck,preferably organic
Salt and freshly ground pepper
1/2 lb mild American blue cheese such as Maytag Blue Cheese, cut into 4 slices
4    large eggs
4    brioche buns,split
4 T  unsalted butter,melted
Vegetable oil
1 c  Green-Chile Relish,warmed up
1. Light a grill. Gently form the beef into 4 thick patties and
season with salt and pepper. Grill over moderately high heat until
lightly charred on the outside and medium-rare within, about 5 minutes
per side. Just before the burgers are done, top each one with a slice
of cheese and let it melt.
2. Meanwhile, heat a large cast-iron skillet and brush it lightly with
oil. Crack the eggs into the skillet and cook sunny-side up over
moderate heat, about 4 minutes. Spread the cut sides of the buns with
the melted butter and toast on the grill. Transfer the burgers to the
buns. Top with the Green-Chile Relish and the fried eggs and serve.

Green-Chile Relish

1 lb long green New Mexico or  Anaheim chiles
3    garlic cloves,minced
Salt and freshly ground pepper
1. Roast the chiles over a gas flame or under a preheated broiler,
turning often, until charred all over. Transfer to a large plate and
let cool. Discard the charred skin, seeds and stems and cut the chiles
into 1/2-inch dice. Transfer the chiles to a bowl and stir in the
garlic. Season with salt and pepper.
Make Ahead The green chile relish can be refrigerated for up to 3 days.