Preheat broiler. Place pepper quarters on a rimmed baking sheet, skin-side up; broil 5 minutes. Rotate pan; broil until skin is blackened and blistered, about 3 to 5 minutes more. Transfer peppers to a bowl; tightly cover. Let peppers steam until cool enough to handle, about 10 to 15 minutes; remove skin with your hands and cut pepper quarters into strips.
Place beans, garlic, lemon juice, oil, cumin, and salt in bowl of a food processor; process until smooth.
Spread about 1/4 cup bean mixture on each of 4 slices toast; top each with 1/4 of roasted pepper, cucumber and spinach. Spread avocado onto remaining slices toast; top sandwiches with toast and serve.