Start by preparing the peach compote so it has time to cool. In a large saucepan over medium heat, add the butter, diced peaches, brown sugar, maple syrup, lemon juice, cinnamon, and salt. Cook until the sugar melts and the peaches start to soften, stirring occasionally (~10 minutes). It can help to steam the peaches by covering the pan with a lid for a few minutes.

Mix together the cornstarch and water in a small bowl to create a slurry. Pour the slurry into the peach mixture, stir, and cook until the mixture thickens (1-2 more minutes).

Remove from the heat and stir in the vanilla. Set aside to cool.

Preheat the oven to 425F. Line a standard 12-cup muffin pan with 6 muffin liners.

In a medium bowl, use a handheld electric mixer to beat the cream cheese to smooth. Then beat in sugar, vanilla, and salt for about 2 minutes, until very smooth and creamy. Transfer half of the cream cheese filling to a piping bag or a large ziplock bag (reserve the other half for after baking). Set aside.

In a medium bowl, mix together the flour, brown sugar, cinnamon, and salt. Then add the butter chunks and mix until combined and crumbly. Set aside.

In a large bowl, whisk together the flour, baking powder, baking soda, salt, and cinnamon. Set aside.

In a separate large bowl whisk together the melted butter, oil, sugar, and eggs until well combined.

Add the buttermilk, Greek yogurt or sour cream, and vanilla. Whisk until well combined.

Add the dry ingredients to the wet, and fold together until just barely combined. Then fold in the diced peaches if using.

Scoop and divide the batter among the prepared muffin cups, filling them to the rim. Spoon about 1-2 tbsp of the peach compote on top of each muffin.

Snip the tip off of the piping bag with the cream cheese filling and place it into the center of the batter for each muffin.

Sprinkle the crumb topping on top of the muffins.

For regular-sized muffins: bake at 425F for 5 minutes, then reduce the oven temperature to 350F and continue to bake for another 15-20 minutes.

For jumbo muffins: bake at 425F for 5 minutes, then turn the oven temperature down to 350F and bake for another 25-30 minutes.

Remove the muffins from the oven and allow them to cool in the pan for 5 minutes.

Pipe the remaining cream cheese filling into each muffin.

Remove them from the pan and transfer to a wire cooling rack.