Preheat oven to 375°F. Line a 9-inch square baking pan with parchment paper, allowing the parchment to extend over the sides by at least 2 inches.
Combine 2 cups strawberries, 1 tablespoon lemon juice and ½ teaspoon vanilla in a small saucepan; bring to a boil, partially covered, over medium-high heat. Reduce to a lively simmer, cover, and cook until the strawberries are starting to break down, about 3 minutes. Mash with a potato masher; stir in 1 tablespoon chia seeds. Cook at a lively simmer, stirring often, until the strawberries are almost completely broken down and the mixture is starting to thicken, 4 to 5 minutes. Remove from heat.
In a large bowl, whisk 2 eggs, ¾ cup banana, ½ cup yogurt, ¼ cup peanut butter, ¼ cup maple syrup, 1½ teaspoons cinnamon, 1 teaspoon baking powder, ½ teaspoon salt and the remaining 2 teaspoons vanilla until smooth. Whisk in ½ cup milk until combined. Fold in 2½ cups oats.
Transfer the oat mixture to the prepared pan. Dollop evenly with the strawberry mixture and the remaining ⅓ cup peanut butter; using a butter knife, gently swirl into oat mixture as desired.
Bake until just set and golden brown, 30 to 35 minutes. Let cool in the pan on a wire rack for 10 minutes before serving.
