Make salsa: Mix all ingredients together until well combined. Set aside.
Heat broiler to high.
Season and cook burger on a grill (grill should be in the 550 degree range to sear meat well); sear sausage to a light char on the outside. Set aside.
Add arugula and olive oil to mixing bowl; toss together. Set aside.
Spread garlic aioli over bottom burger bun and then top with baby arugula mix. Add burger and layer four sausage slices over burger and top with salsa criolla. Top salsa with Monterey Jack cheese slice, melt under broiler.
While cheese is melting, fry egg in a medium saucepan over medium high heat until just set. Slide egg gently onto burger, top with bun. Serve immediately.