In a Dutch oven over medium-high heat melt 1 tablespoon each of the butter and olive oil. Add onions and cook, stirring occasionally, 5 minutes, then reduce heat to medium.
Cook until onions are caramelized and rich, deep golden brown, about 25 minutes, stirring often. Turn heat to medium-low or add water if onions are dry or start to scorch before caramelizing.
Add brown sugar, balsamic vinegar, salt, pepper, and red pepper flakes; cook 1 minute. Onions should be jammy and darkly caramelized when done. Remove to a plate to cool.
To make sandwiches:
Place bread slices on a flat surface. Spread a thin layer of mayonnaise on one side of each bread slice.
Turn bread slices over and spread goat cheese on plain sides. Top goat cheese with Gruyère, an even layer of beets, caramelized onions, and arugula. Top with remaining bread, mayo sides out. Secure with toothpicks if desired.
In a large skillet over medium-low heat melt remaining butter with olive oil. Add two sandwiches; cover and grill until bottoms of bread are toasted, about 5 to 7 minutes, and cheese starts melting, checking often. Flip and cook, uncovered, until cheeses are melted and bottoms of bread are toasted.
Repeat with the remaining sandwiches. Makes 4 sandwiches. Recipe from Seneca Foods.