"I love this as a fancy alternative to plain grilled cheese, and the aioli is an instant hit. Spread the sauce on top of the bread for a messier, fork-needed meal. You'll have enough aioli for a second sandwich; it will harden in the fridge due to the olive oil, but softens quickly. It really only lasts a day."
Combine garlic and lemon juice in a bowl; mix egg yolk in gently. Add olive oil, dill, salt, and cayenne; stir aioli until combined.
Heat a large skillet over medium heat. Butter 1 side of each bread slice; place 1 butter-side-down into the skillet. Layer 2 slices Cheddar cheese, mushrooms, and spinach on the bread. Cover with 2 remaining slices of Cheddar cheese and the remaining slice of bread, butter side up. Fry sandwich until dark brown and crispy, about 5 minutes per side. Serve with aioli for dipping.