Preheat oven to 350 degrees Fahrenheit. Spray an 8 by 8 square pan with nonstick cooking spray.
First make your crumb topping. Using your hands, a food processor or a pastry cutter mix the cold butter into the flour until it is crumbly and butter is completely combined into the flour. Set aside.
In a medium saucepan add frozen berries, lime juice, water and sugar on medium high heat. Stir and bring to a boil. Once boiling turn down to medium heat and let simmer for 10 minutes. At this point the mixture should have thickened. Turn heat to medium low and cook for 7 more minutes, stirring occasionally. Take off the heat and let cool.
Meanwhile, In a stand mixer beat butter, salt and both sugars for 4 minutes on medium high until light and fluffy.
Measure flour ingredients in a small bowl and stir together with a small spatula or fork. Measure milk and yogurt together then set aside.
Add each egg to the butter mixture one at a time, scraping the bowl with a spatula after each is incorporated.
Add one third of the flour mixture to the mixer and beat until incorporated. Do the same with one third of the milk mixture. Continue to do so until all the milk and flour is incorporated and the batter looks thick and smooth.
Add half of the batter to the greased 8 by 8 pan then spread the berry mixture evenly over the top. Cover evenly with the rest of the batter. Add your crumb mixture evenly over the top.
Bake for around 55 minutes until a sharp knife or toothpick comes out clean.