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For the Chimichurri, place parsley, garlic, oregano, fennel seed and cayenne pepper in food processor; cover. Pulse until garlic is chopped. With food processor running, gradually add oil and lemon juice. Stir in sea salt. Transfer 1/2 cup of the Chimichurri to large bowl. Reserve remaining Chimichurri until ready to serve.
For the Burgers, add ground beef to Chimichurri in bowl. Mix until well blended. Shape into 4 patties.
Grill over medium heat 4 to 6 minutes per side or until burgers are cooked through (internal temperature of 160°F). Add cheese slices to burgers 1 minute before cooking is completed. Toast rolls on the grill, open-side down, about 30 seconds.
Serve burgers on rolls with red onion and reserved Chimichurri.