Brown the sausage in a skillet; transfer to a paper-towel-lined plate and set aside.
Heat salsa in a saucepan over medium heat until warm. Stir in beans and cilantro; keep warm over low heat.
Whisk together 3 eggs and 1 Tablespoon sour cream for the first omelet. Pour egg mixture into the skillet and season with salt and pepper. Immediately begin gently shaking the skillet back and forth while making small, quick stirring motions. Reduce heat to low when eggs no longer run but still are moist.
Add sausage and pepper Jack cheese filling in a line.
To serve, spoon sauce on serving plate. Fold 1/3 of the omelet over the filling, then slide the exposed third of the omelet from the edge of the pan onto a serving plate, folding the rest of the omelet over on the plate.
Garnish with a dollop of sour cream and cilantro. Repeat to make 3 more omelets.