Preheat oven to 375°F with rack in middle. Butter a 9-inch square cake pan.
Stir together flour, baking soda, and salt.
Beat together softened butter (1 stick) and 3/4 cup sugar in a medium bowl with an electric mixer at medium speed until pale and fluffy, then beat in eggs 1 at a time until blended. Beat in bananas, yogurt, and vanilla (mixture will look curdled).
With mixer at low speed, add flour mixture and mix until just incorporated.
Toss together chocolate, nuts, cinnamon, melted butter, and remaining 1/4 cup sugar in a small bowl. Spread half of banana batter in cake pan and sprinkle with half of chocolate mixture. Spread remaining batter evenly over filling and sprinkle remaining chocolate mixture on top.
Bake until cake is golden and a wooden pick inserted in center of cake comes out clean, 35 to 40 minutes. Cool cake in pan on a rack 30 minutes, then turn out onto rack and cool completely, right side up.