In a small mixing bowl, use a fork to stir the cream cheese and sugar together until smooth. Refrigerate mixture until ready to use.

In a small saucepan over medium heat, bring the raspberries and 1 Tablespoon sugar to a simmer while using a fork to mash the berries. Continue stirring until the berries have broken down and the mixture has thickened slightly. Pour mixture through a fine mesh strainer set over a small bowl and cool to room temperature.

In a small bowl, combine the milk and lemon juice. Set aside.

In a large mixing bowl, whisk together the flour, baking powder and salt. In a stand mixer fitted with a paddle attachment, beat the eggs, brown sugar, and remaining 3/4 cup granulated sugar on medium until smooth and pale in color, about 2 minutes. Add the greek yogurt, vanilla, and lemon-milk mixture and continue beating until combined. With the mixer on low, slowly stream in the butter and vegetable oil.

Pour wet mixture into flour mixture and use a rubber spatula to gently fold everything together until just combined. Try to avoid over-mixing which will produce tougher, dry muffins.

Fill muffin cups 1/2 full with batter and top with about a teaspoon of the cooled raspberry mixture. Evenly distribute the remaining batter among the muffin cups and smooth the tops. Top muffins with another teaspoon of the raspberry mixture and use a toothpick to gently swirl the berry mixture throughout the batter.

Spoon 1 to 2 teaspoons of the cream cheese filling on top of each muffin cup and use a knife or offset spatula to gently press mixture further into the batter. Muffin cups may look rather full at this point, but they WILL bake up fine! Sprinkle with coarse sugar.

Bake for 15-17 minutes, or until muffin tops are lightly golden and springy to the touch. Allow muffins to cool in the pan for 5 minutes before transferring to a wire rack to cool completely. Store leftover muffins covered in the refrigerator.