For the jalapeno chimichurri: In a food processor, combine the parsley, cilantro, jalapeno, garlic and vinegar. Process until all ingredients are well chopped and combined. Add the mayonnaise and pulse several times to incorporate. Set aside.
For the panini: Spray a skillet with cooking spray and place over medium heat. Fry the eggs until set, about 2 minutes on each side. Set aside.
Spread 1 tablespoon of the chimichurri on both slices of each slice of bread. Top half the slices with 2 slices of Brie, a fried egg, some avocado, and 2 more slices of Brie. Place the remaining slices of bread on top. Spread about 1 teaspoon of butter on the outside of each sandwich.
Heat a large cast-iron skillet over medium heat. Place 2 sandwiches at a time in the skillet (don¿t overcrowd the pan). Cook until the bread is golden brown and the cheese begins to melt, 2 to 3 minutes. Flip and cook 2 to 3 more minutes to brown the other side. Repeat with the remaining sandwiches. Slice the sandwiches in half and serve hot.