A traditional kaiserschmarrn is torn or cut after baking and tossed with compote. At her Los Angeles restaurant République, chef Margarita Manzke serves her version whole, with the compote baked right in to layer the flavor into each bite.
Preheat oven to 375°F with oven rack in lower third of oven. Combine butter and 1/4 cup granulated sugar in a medium skillet over medium. Cook, stirring constantly, until sugar starts to dissolve, butter is melted, and mixture is lightly browned, 3 to 5 minutes. Add apple, lemon juice, and 1/4 teaspoon salt. Cook, stirring constantly, until apples are crisp-tender and sauce is syrupy, about 2 minutes. (Sauce may seize up when you add the apples but will smooth out as it cooks.) Remove skillet from heat; turn off burner.
Pour 2 tablespoons rum into a heatproof measuring cup with a pour spout. Working quickly, pour rum into skillet with apple mixture. Carefully ignite the fumes just above mixture with a stick lighter or long match. Gently shake skillet until flames extinguish, about 20 seconds.
Place skillet over medium-high, and cook, stirring often, until mixture is syrupy, about 1 minute and 30 seconds. Pour apple mixture through a fine wire-mesh strainer into a bowl, reserving syrup in a bowl for drizzling. Cool apple slices completely, about 15 minutes, and set aside.
Whisk together egg yolks, 1/4 cup granulated sugar, and remaining 1 tablespoon rum in a large bowl. Whisk in cream cheese and sour cream until mostly smooth. Whisk in flour and remaining 1/4 teaspoon salt until smooth. Pour batter through a fine wire-mesh strainer into a large bowl, pressing mixture with the back of a spoon to remove any lumps. Set aside.
Beat egg whites in bowl of a heavy-duty stand mixer fitted with a whisk attachment on medium speed until frothy, about 30 seconds (reserve yolks for another use). Add cream of tartar. Continue beating on medium speed, gradually adding remaining 1/4 cup granulated sugar, until mixture is glossy and forms soft peaks, 2 to 3 minutes (do not overwhip). Working in 2 batches, gently fold egg white mixture into batter mixture until just combined.
Grease a 10-inch ovenproof skillet with butter; dust with granulated sugar. Arrange flambéed apple slices in an even layer on bottom of skillet. Top with batter, smoothing top. Bake in preheated oven until golden brown, puffed, and set, 28 to 30 minutes, tenting with aluminum foil after 22 minutes if needed to prevent excess browning. Top with ice cream, and dust with powdered sugar. Serve immediately with reserved flambéed apple syrup.
Apples and syrup mixture can be made 1 day ahead.
Refrigerate leftover egg yolks in an airtight container up to 3 days.