Line the pan with parchment and heat the oven to 360 °F
Whip 3 medium eggs and ½ cup granulated sugar with the wire whisk until pale and fluffy, about 5 minutes. On low speed, mix in 3 Tablespoons neutral oil and 3 Tablespoons water.
Sift ¾ cup all-purpose flour, 3 Tablespoons dark cocoa powder, 1 ¼ teaspoon baking powder, and 1 pinch salt. Mix into the egg foam on the lowest speed (or fold by hand), a third at a time, just until no streaks remain. Spread evenly in the pan.
Bake 10 to 13 minutes, until a toothpick comes out clean. Cool completely in the pan.
With the flat beater, cream 1 ½ sticks unsalted butter and ¾ cup powdered sugar until light, 5 to 8 minutes. Beat in 1 ⅓ pounds tvaroh cheese a spoonful at a time, then 1 teaspoon vanilla extract and 1 Tablespoon rum. Beat just until smooth.
Spread the cream over the cooled base. Chill 30 minutes (or 15 to 20 minutes in the freezer).
Bring ½ cup heavy cream just to a boil. Off the heat, add 5 ounces semisweet chocolate; let stand 1 to 2 minutes, then stir gently until smooth. Cool 5 to 10 minutes to thicken.
Pour the ganache over the cream and smooth. Chill 2 hours or overnight, until set.
Cut into squares with a knife dipped in hot water and wiped dry between cuts.
Notes
Makes about 12 squares in a 9×13-inch (23×33 cm) pan.
Oven: temperatures are for a conventional (non-fan) oven; for convection, lower by 25°F.
Tvaroh: use firm, full-fat tvaroh or farmer's cheese, at room temperature. Not cream cheese or a soft/low-fat cheese, or the cream layer will not slice.
Have the butter and tvaroh at room temperature; do not overbeat the cream (it can split).
Chocolate: use a chopped baking bar, not chips.
Make-ahead: best a day ahead; it firms up overnight.
Storage: keep in the pan, covered, refrigerated, up to 4 days.