Using a sharp knife, carefully form a pocket in the side of each bread slice. Avoid cutting completely through the bread.
Place the softened cream cheese, pumpkin purée, brown sugar, 1 teaspoon vanilla extract, and pumpkin pie spice in a medium bowl. Stir until the mixture is thoroughly combined, smooth, and creamy.
Divide the pumpkin cream cheese mixture among the prepared bread slices. Spoon the filling gently into each pocket, then press the bread lightly so the mixture spreads evenly without tearing the slices.
In a large mixing bowl, whisk together the beaten eggs, melted butter, whole milk, 2 teaspoons vanilla extract, granulated sugar, all-purpose flour, and salt. Continue whisking until the batter is uniform and free of lumps.
Dip each filled bread slice into the batter, coating both surfaces thoroughly.
Warm a skillet over medium-high heat. Add a small quantity of butter or oil, if desired.
Arrange the coated bread slices in the heated skillet. Cook each side for 2–3 minutes, or until the exterior is golden brown and the French toast is heated through.
Transfer the finished French toast to serving plates and serve immediately while warm.