Cut Monterey Jack cheese into 3- x 1/4- x 1/4-inch sticks.
Cut a lengthwise slit in each okra pod, cutting to but not through ends; push seeds aside. Stuff pods with cheese sticks, and set aside.
Combine flour and cornmeal in a large bowl; make a well in center of mixture.
Stir together egg, buttermilk, and beer; add to dry ingredients, stirring until smooth.
Pour oil to depth of 3 inches into a Dutch oven; heat to 375°. Dip stuffed okra in batter, coating well; fry, a few at a time, in hot oil until golden. Drain on paper towels. Sprinkle with salt; serve immediately with salsa.