Place an oven rack in the center position and preheat the oven to 350°F (175°C). Line two rimmed baking sheets with parchment paper or silicone baking mats and confirm all ingredients are at room temperature.
Beat 1/2 cup softened butter with 4 ounces softened cream cheese on medium speed for about 1 minute until smooth. Add the granulated sugar and brown sugar and beat 2 to 3 minutes until pale and fluffy, scraping down the bowl.
Add the whole egg, egg yolk, and 2 teaspoons vanilla extract and beat on low speed for about 30 seconds until just combined. Do not overbeat, as too much air can make the cookies puff up and collapse.
Whisk together the flour, baking soda, baking powder, and 1/2 teaspoon salt in a medium bowl. Add to the wet mixture in 2 or 3 additions on low speed, stopping as soon as no streaks of flour remain. The dough will be soft and slightly sticky.
Fold in the white chocolate chips with a rubber spatula until evenly distributed. Cover the bowl and chill the dough in the refrigerator for 30 minutes to firm it up and prevent excess spreading.
Scoop the chilled dough into balls of about 1 1/2 tablespoons each and place them about 2 inches apart on the lined baking sheets, aiming for about 12 cookies per sheet. If the dough feels very soft, chill the trays for 10 minutes before baking.
Bake one sheet at a time on the center rack for 8 minutes, until the edges are just set and the tops look puffy. Press the back of a 1/2 teaspoon measuring spoon into the center of each cookie to make a well, then bake 3 to 4 more minutes until the edges are lightly golden and the centers still look soft.
Let the cookies rest on the baking sheet for 5 minutes, then move them to a wire cooling rack and cool for at least 30 minutes until fully cool to the touch. Do not frost warm cookies or the frosting will melt and slide off.
Beat the remaining 4 ounces softened cream cheese with 2 tablespoons softened butter until smooth. Add the powdered sugar, 1/2 teaspoon vanilla, and a small pinch of salt, and beat 1 to 2 minutes until light, fluffy, and spreadable. Transfer to a piping bag or zip-top bag and snip off the corner.
Stir the raspberry jam well so it loosens up. Pipe a generous swirl of frosting into the well of each cookie, spoon about 1 teaspoon of jam over the frosting, and top with a fresh raspberry half. Serve right away or chill for 15 minutes to set the frosting.