For the burgers: Heat 1/4 cup of the oil over medium-high heat in a 12-inch nonstick skillet. Add the shallots, 3/4 teaspoon salt and 1/4 teaspoon pepper. Cook until soft, about 3 minutes. Add the mushrooms, garlic, thyme and the remaining 3/4 teaspoon of salt and 1/4 teaspoon pepper. Cook until the mushrooms are soft, 6 to 8 minutes. Set aside to cool slightly.
Blend the peas and 1 can lentils until smooth in a blender. Transfer the pureed peas and lentils to a medium bowl. Add the remaining 1 can lentils, 1/3 cup cornmeal, mayonnaise, lemon juice and the mushroom mixture. Form 1/3-cupfuls of the mixture into eleven 3/4-inch-thick slider patties.
Sprinkle 1/4 cup cornmeal on a baking sheet. Place the burgers on top of the cornmeal. Sprinkle the remaining 1/4 cup cornmeal on top of the burgers. Refrigerate for at least 30 minutes.
Heat the remaining 1/4 cup oil over medium heat in the skillet. Cook the burgers until golden brown, about 4 minutes on each side.
For the mayonnaise: Mix the mayonnaise, basil and lemon zest until smooth in a small bowl.
Working in batches, place the bread in the skillet and cook until lightly toasted, 1 to 2 minutes on each side. The bread can also be grilled or toasted in an oven.
Smear the mayonnaise on the bread and top with the lentil burgers. Serve with avocado if using, tomato, lettuce and top with the remaining the bread slices.