Combine the cup of berries and teaspoon of sugar in a small saucepan. Cook and stir over medium heat for about 8 minutes, or till berries are mashed and the mixture has thickened. Remove from the heat and let cool while you prepare the muffins.
Preheat your oven to 425° and spray 18 muffin cups with cooking spray (or use paper liners).
Crumb Topping:
Combine the sugar, flour, and softened butter in a small bowl. Use your hands or a fork to mix until crumbly. Set aside.
Batter:
Stir or whisk together the flour, baking powder, and salt in a small bowl; set aside.
In a large bowl, whisk the eggs and sugar for at least 2 minutes, or until pale and thickened. Add the melted butter, oil, buttermilk, and vanilla.
Add the flour mixture to the liquid ingredients, then dump the blueberries onto the flour. Stir gently just until combined.
Fill muffin cups about 2/3 full. Spoon about 1 tsp of cooked berries on top of each muffin. Swirl in with a butter knife.
Combine topping ingredients till crumbly and sprinkle on top of muffins.
Bake muffins at 425° for 10 minutes. Turn the oven temperature down to 350° and bake for another 12-15 minutes or until the edges are golden brown and a toothpick inserted in the middle comes out without wet batter.
Let the muffins cool in the pan for 5-7 minutes, then carefully remove to a wire rack.
Notes
For lemon blueberry muffins, add a tablespoon or two of lemon zest to the batter, and 1/2 teaspoon of lemon zest to the crumb topping.
You can use fresh or frozen berries for the compote, but for the batter, I highly recommend using fresh blueberries. Frozen berries prevent the muffins from cooking in the middle before the outside is done, and thawed berries turn the muffins an ugly gray color.
If you don't have buttermilk, you can use 1/2 cup milk + 1/2 cup sour cream or plain yogurt.
Muffins can be stored at room temperature in an airtight container for 3-5 days. You can also freeze them in a heavy-duty Ziploc freezer bag for up to 3 months. To prevent the crumb topping from falling off, I like to flash-freeze them first on a cookie sheet, then wrap them individually in plastic wrap before placing them in the bag.