Preheat oven to 180 degrees. Place the beets on a baking sheet and bake for 45 minutes. Cool, peel, and slice thinly.
In a bowl, combine cream cheese, chives (leaving some for garnish), lemon juice, and shallots; mix well.
Cut each slice of bread into square pieces (24 for 12 sandwiches). Fry until golden brown. Cool completely.
Spread a thin layer of cheese mixture on a square of bread. Spread the trout and beets evenly on top; cover with another square. Garnish with beets and onions.