Begin by browning half of the butter. In a medium-sized, light colored saucepan over medium heat, melt 1/2 cup (113g) of the butter and cook until it's deep golden and fragrant. When brown specks begin to appear, remove it from the heat and immediately transfer the butter to a small bowl. Let it sit until it reaches room temperature, then refrigerate it until firm.
Preheat the oven to 350ºF (180ºC). Line 2 large cookie sheets with parchment paper.
In a large bowl, beat the butter, brown butter, brown sugar and granulated sugar together, with an electric mixer on medium speed, until pale and fluffy. Beat in the eggs in 2 additions, making sure each one is incorporated, then mix in the vanilla extract. On low speed, stir in the flour mixture in 2 additions, just until combined, then mix in the white chocolate pieces.
Set aside 3 tablespoons (54g) of the cooled strawberry jam. Add dollops of the remaining strawberry jam to the cookie dough. Stir very gently, but don't mix in the jam completely. You want to see strawberry swirls in the dough. Do not over mix! Over mixing will change the texture of the cookies.
Place 3-tablespoon sized portions of cookie dough about 3 inches apart onto the prepared cookies sheets. (I used a #24 portion scoop.*) Top each cookie with some of the crumb topping, pressing lightly to adhere. Top each one with about a half teaspoon of the reserved strawberry jam.
Bake the cookies until the edges are golden brown and the tops are set, about 12-14 minutes. If you want to make your cookies more perfectly round, immediately use a large round cookie cutter to shape the hot cookies. Place the cookie cutter on the cookie sheet over each cookie and slide it around a couple of times in a circular motion. Let the cookies cool on the pan for 1 minute, then transfer them to a cooling rack.