Heat ½ tablespoon oil in a large nonstick skillet over medium heat. Add onion, and saute 4 to 5 minutes or until tender and golden. Add garlic, and saute 30 seconds longer. Let cool.
Combine 1 tablespoon milled flaxseed and 3 tablespoons water in a small bowl, stirring with a fork to combine. Let stand for 2 to 3 minutes or until the mixture thickens to a gel-like consistency.
Combine flaxseed mixture and beans in a large bowl, mashing with a potato masher until combined. Stir in sauteed onion, quinoa, oats, ¼ cup milled flaxseed, the Italian seasoning, salt, and pepper. (Refrigerate mixture for 30 minutes for easier shaping.) Shape mixture into 4 patties.
Heat remaining 1 tablespoon oil in skillet over medium heat. Cook 3 to 4 minutes on each side or until crisp and heated through.
Spread 1 tablespoon pesto over the bottom half of each sandwich thin; add patties. Top evenly with spinach leaves, tomato slices and sandwich thin tops.