Using a large bowl, add cheese and cornstarch; toss until well combined.
Transfer mixture into medium saucepan.
Add evaporated milk, smoked paprika and salt; stir and place over medium-low heat.
Cook, stirring frequently, for 3-4 minutes, or until sauce is smooth and has thickened noticeably.
Reduce heat to low to keep warm until needed.
For the Mushrooms
Using a large skillet, add olive oil and place over medium-high heat.
Once hot, add onions, mushrooms, salt and pepper; sauté for 5 minutes, stirring occasionally.
Reduce heat to medium-low. Add butter and garlic; continue sautéing for 4-5 minutes, or until mushrooms and onions have softened.
Remove skillet from heat.
For the Burgers
Preheat grill to medium-high heat.
Using a large bowl, mix all of the burger ingredients until just combined. Shape into 6 patties and gently indent the center of each patty with your thumb.
Brush grill grates with vegetable oil or spray with nonstick grilling spray. Place burgers on the grill; grill for 4-5 minutes per side, or until burgers reach an internal temperature of 160°F.
{Optional} Place burger buns cut-side down on grill for 1-2 minutes.
Begin assembling burgers by layering lettuce, tomato and patties on top of the buns. Add cheese sauce on top and then top with sautéed mushrooms and onions.