This cake can be baked on the grill or in the oven.
Make ahead: The optional herbed crème fraîche needs to rest in the refrigerator for at least 30 minutes and up to overnight. Adapted from "Michael Chiarello's Live Fire: 125 Recipes for Cooking Outdoors" (Chronicle Books, $35, 224 pages) with Ann Krueger Spivack and Claudia Sansone.
For the herbed crème fraîche:
1 cup crème fraîche
1 1/2 teaspoons minced fresh rosemary (may substitute minced fresh basil leaves)
For the cake:
6 tablespoons (3/4 stick) unsalted butter, at room temperature, plus more to grease the pan
1 1/2 cups flour
1 1/2 teaspoons baking powder
1/2 teaspoon coarse sea salt, preferably sel gris
1/2 cup granulated sugar
1/4 cup packed dark brown sugar
1 large egg
1/2 cup whole or low-fat milk
1 teaspoon vanilla extract
1 pound strawberries, hulled and halved
For the balsamic glaze:
1/4 cup balsamic vinegar
1 tablespoon plus 1 teaspoon granulated sugar
A few grinds of fresh black pepper
Confectioners' sugar, for dusting (optional)
Roasted strawberries with syrup (optional; see note)
For the crème fraîche:
Stir together the crème fraîche and rosemary in a medium bowl. Cover and refrigerate for at least 30 minutes or overnight.
For the cake: Butter a 9-inch cast-iron skillet or an 8-by-8-inch enameled cast-iron baking pan (if cooking in the grill) or a standard 8-by-8-inch baking pan (if baking in your oven). Don't put a standard baking pan inside your grill or over a fire. Light the coals, turn a gas grill to medium, or preheat an oven to 350 degrees.
Whisk together the flour, baking powder and salt in a mixing bowl.
Combine the butter and sugars in the bowl of a stand mixer or hand-held electric mixer. Beat on medium speed for 3 minutes, until creamy. Stop to scrape down the sides of the bowl.
On medium speed, gradually add the egg, milk and vanilla extract, beating until just combined. The mixture might begin to look curdled; that's OK. Reduce the speed to low and gradually add the flour mixture, mixing just until smooth and creamy.
Pour into the pan, using a flexible spatula to smooth and level the surface. Arrange the strawberry halves, cut side down, on the batter. Don't overlap the berries; use just enough for one layer and reserve any extras as a garnish, if desired.
To bake in a grill: When the grill reaches 350 degrees, slide in the cake, resting the pan on the grill rack; close the grill's lid and let it bake for at least 20 minutes. Test the cake: It's done when a skewer inserted into the center (but not in a strawberry) comes out clean and the cake is golden brown on top.
To bake in an oven: Bake for 10 minutes, then reduce the heat to 325 degrees and bake for 45 to 50 minutes. The cake is done when a skewer inserted into the center (but not in a strawberry) comes out clean and the cake is golden brown on top.
For the balsamic glaze: While the cake is baking, combine the balsamic vinegar, sugar and black pepper in a small saucepan. Bring to a boil over high heat, then reduce the heat to medium and allow the mixture to boil for 2 minutes, stirring frequently. It should be the consistency of maple syrup; if it thickens too much before the cake comes out of the oven, stir in more vinegar as needed.
As soon as the cake comes off the heat, drizzle about three-quarters of the balsamic glaze over the top.
When the cake has cooled, cut it into wedges if baked in a skillet; if baked in a standard baking pan, cut it into 8 squares.
To assemble: Either garnish simply with a light dusting of confectioners' sugar and garnish with any remaining fresh strawberry halves; or to dress up the cake, spoon some roasted strawberries and their juices, or any remaining fresh strawberry halves, onto one side of each plate; on the other side of the plate, pour a small pool of the balsamic glaze. Drizzle more balsamic glaze over the cake if you like. Set a cake slice in between and top with a spoonful of herbed crème fraîche. Spoon any remaining crème fraîche into a bowl and set on a platter with leftover halved strawberries to pass at the table.
Per serving (using whole milk): 400 calories, 5 g protein, 46 g carbohydrates, 22 g fat, 14 g saturated fat, 80 mg cholesterol, 210 mg sodium, 2 g dietary fiber, 24 g sugar