Preheat the oven to 375°F (190°C). Line a 12-cup muffin tin with paper liners.
In a large bowl, whisk together the all-purpose flour, baking powder, salt, and 1 cup granulated sugar until well combined.
In a separate medium bowl, beat the eggs. Add the whole milk, vegetable oil, fresh lemon juice, lemon zest, and vanilla extract. Whisk until smooth.
Pour the wet ingredients into the dry ingredients and gently fold together using a spatula until just combined. Do not overmix; the batter should be slightly lumpy.
Spoon about 1 tablespoon of batter into each muffin cup, filling them about one-third full.
Using a small spoon or piping bag, add about 1 teaspoon of lemon curd into the center of each muffin cup on top of the batter.
Top the lemon curd with the remaining batter, filling each muffin cup about two-thirds full.
Bake for 18 to 20 minutes or until the muffins are golden and a toothpick inserted into the muffin (not the lemon curd) comes out clean.
Remove the muffins from the oven and allow them to cool in the pan for 5 minutes before transferring to a wire rack to cool completely.
To make the lemon curd: In a medium heatproof bowl, whisk together the softened unsalted butter, 1 cup granulated sugar, 3 eggs, 1/2 cup fresh lemon juice, lemon zest, and salt.
Place the bowl over a pot of simmering water (double boiler) making sure the bottom of the bowl does not touch the water.
Whisk continuously for about 10 minutes until the mixture thickens and coats the back of a spoon.
Remove from heat and strain the lemon curd through a fine mesh sieve into a clean container to remove any lumps.
Allow the lemon curd to cool to room temperature before using it in the muffins or storing it in the refrigerator.