Prepare the jalapeno pesto. Preheat oven to 400 degrees F. Place the jalapeno on a baking sheet and roast for 10 minutes or until skin begins to blister and darken. Remove from heat and transfer jalapeno to a small bowl. Cover with plastic wrap and let sit for 10 minutes. Remove cover and carefully peel the outer skin off the jalapeno.
Blend together roasted jalapeno with basil, garlic, toasted pine nuts, Parmesan, and olive oil. Season with salt and pepper. Set aside.
Prepare the caramelized onion. Heat oil in a medium saute pan over medium heat. Add onions and season with salt. Continue to cook, stirring occasionally, until onions are deep brown and caramelized, about 20 minutes. Add 2-3 tablespoon water if the pan becomes too dry. Remove from heat and set aside.
Reduce oven temperature to 275 degrees F. Place a metal wire rack on a baking tray.
Combine seasonings for ribeye steaks in a small bowl. Season both sides of the steaks with the seasoning mixture. Place steaks on the metal wire rack and roast until the internal temperature comes to 110 degrees (depending on the thickness of steaks, this can take anywhere between 30-60 minutes). Remove steaks from the oven and let rest for 10 minutes.
Meanwhile, prepare the cheese sauce. Melt butter in a medium saucepan over medium heat. Add flour and whisk until combined. Add milk, whisking constantly. Let the milk come to a simmer, stirring occasionally, and reduce heat to low. Add garlic powder and half of the Monterrey jack and white cheddar cheese. Stir until the cheese melts and add the remaining amount. Stir until melted and the sauce is smooth about 5 minutes. Season with salt and pepper. Keep warm.
Heat 1 tablespoon butter in a saute pan over high heat. Add steaks and sear until browned, about 1 minute. Flip and cook on the other side another minute or until browned. Remove from heat. Slice steaks across the grain.
Assemble sandwiches. Toast sandwich buns. Spread jalapeno pesto on the bottom bun, top with arugula, ribeye, cheese sauce, and caramelized onions. Serve immediately.