In a bowl, stir Neufchâtel cheese until smooth, using a whisk or the back of a spoon to help get rid of lumps. Mix crème fraîche, honey, and lemon zest into Neufchâtel. Fold in 1 1/2 cups strawberries.
Spread berry mixture onto 4 slices of bread, leaving about 1/4-in. border. Top with remaining plain slices of bread, pressing down lightly.
Thoroughly whisk eggs, milk, vanilla, and salt in an 8- or 9-in. baking dish. Whirl pistachios in a processor to finely chop, then spread on a plate.
Quickly dip 1 French toast stack into custard, making sure all sides and edges are covered, then set it in pistachios, turning to coat both flat sides. Set on a plate. Repeat to coat remaining French toast stacks. Sprinkle about 1/2 tsp. sugar on top side of each stack.
Heat oil and butter in a nonstick 12-in. frying pan over medium heat. Flick a drop of water into pan. If it sizzles, pan is ready for cooking.
Add French toast stacks to pan, sugar side down. Sprinkle top of each stack with 1/2 tsp. sugar. Cook, turning once, until dark golden brown on each side, 3 to 5 minutes per side.
Cut French toast stacks in half diagonally. Set on plates and top with remaining 1/2 cup strawberries and the syrup.
*If you have only fresh bread, put slices on a rimmed baking sheet and bake in a 200° oven until dry to touch, about 20 minutes, turning halfway through baking.