Make the cilantro-avocado cream sauce: In a blender, combine the cilantro, avocados, sour cream, lime juice and a pinch of salt. Blend until smooth. Set aside.
Make the beef filling: Heat the oil in a large skillet over medium heat. Add the onions, salt, a few grinds of pepper, the cumin and thyme. Cook, stirring, until the onions are tender, about 5 minutes. Add the garlic and cook until fragrant, about 5 minutes. Add the beef and break it up with a spoon or spatula as it browns. When the meat is cooked through, taste and, if needed, season with a pinch of salt.
Fill the chimichangas: Warm the tortillas in a microwave for just a few seconds. Using a rubber spatula or the back of a spoon, spread one-quarter of the beans in the center of each tortilla, making a rectangle with the rough dimensions of 6 inches by 3 inches. Add the beef directly over the beans. This leaves 2 inches of tortilla on the rectangle's short sides of the filling and 3 1/2 inches of tortilla on the long sides.
Roll the chimichangas: With the length of the rectangle of fillings horizontal to you, use both hands to fold each side in and over the mound of beef. Then fold the flap closest to you over the beef, tuck it in gently but tightly and roll the burrito away from you. Set aside with the seam side down and continue with the rest.
Fry the chimichangas: Fill a large straight-sided pan with an inch of oil and bring it to 350 degrees F. When the oil is hot, add 2 burritos at a time, seam-side down, adjusting the temperature to keep the oil at 350 degrees F. Cook until golden on the bottom, then roll and cook all sides, about 3 to 5 minutes total. Transfer to a paper towel-lined plate. Repeat with the rest.
Top the chimichangas: Turn on the broiler or preheat your oven to 500 degrees F. Place the chimichangas on a baking sheet, seam-side down. Mix the cheddar and jack cheeses in a bowl. Top each chimichanga evenly with cheese. Cook until the cheese melts, about 1 minute. Transfer each to a serving plate and top evenly with lettuce, cilantro-avocado cream sauce, tomatoes, pickled jalapenos and radish slices.
Photograph by David Malosh
Recipe courtesy Sunny Anderson for Food Network Magazine