In a medium saucepan, melt butter over medium heat. Reduce heat to medium-low. Continue to cook, without stirring, for 5 to 6 minutes or until butter becomes brown and fragrant. Remove from heat; cool slightly. Transfer to a bowl. Cover and chill for 2 hours or freeze for 30 minutes or until firm.
Preheat oven to 325 degrees F. Grease and flour a 10-inch fluted tube pan; set aside.
For filling, in a bowl, combine 3/4 cup of the pecans, 1/2 cup of the brown sugar, and the 2 teaspoons flour. Add 3 tablespoons of the browned butter and work in with fingers or a fork until mixture is crumbly; set aside. In another bowl, stir together the 3 cups flour, the baking powder, baking soda, and salt; set aside.
In a mixing bowl, beat remaining browned butter with an electric mixer on medium to high speed for 30 seconds. Add remaining 1- 1/2 cups brown sugar; beat until combined, scraping bowl occasionally. Add eggs one at a time, beating after each addition until combined. Stir in vanilla. Alternately add flour mixture and yogurt to butter mixture, beating on low speed after each addition just until combined. Stir in the remaining 1-1/4 cups pecans. Spoon half of the batter into prepared pan, spreading evenly. Sprinkle evenly with filling. Spoon remaining batter over filling, spreading to cover.
Bake about 50 minutes or until a wooden toothpick inserted near the center comes out clean. Cool in pan on a wire rack for 10 minutes. Remove from pan. Cool completely on wire rack. Drizzle with Coffee Icing.
In a bowl, stir together 4 teaspoons milk and instant coffee crystals until dissolved. Stir in powdered sugar and enough additional milk (1 to 2 teaspoons) to make a drizzling consistency.