Preheat the oven to 350 degrees F. Mix together the ingredients for the cobbler crumbs until they resemble crumbs. Place on a parchment-lined baking sheet and bake for about 15 - 20 minutes, stirring after the first 10 minutes. Remove and break apart. Allow to cool.
Bring the oven to 425 degrees F. Line a muffin tin with liners, or grease well.
Take the 4 tablespoons of sugar and one cup of the berries and heat on medium-high in a saucepan. Mash the berries as they cook down and continue to cook until the mixture thickens and starts to look like jam. It should still be somewhat liquidy since the mixture thickens as it cools. Set aside and allow to cool completely.
In a large bowl, mix together the flour, baking powder, and salt. Set aside.
In another bowl, mix together the remaining sugar and the eggs until it becomes a pale yellow and the mixture ribbons when you pull up your whisk from the bowl. Mix in the oil, and then the buttermilk and extract.
Gently fold in the wet ingredients into the dry until just combined. It should still be quite lumpy. Divide the batter into the muffin cups and gently swirl about a teaspoon of the berry jam we made earlier. Top the muffin with more of the berries and gently push them in a bit so they stick to the top of the muffins.
Bake for 5 minutes, then without opening the oven door, bring down the heat to 350 and continue to bake for another 12 – 15 minutes, or until a golden brown. While the muffins are still warm, but not hot, brush with the remaining jam.