Melt 4 tbsp. butter in a 4-qt. saucepan over medium-high heat.

Add ½ cup granulated sugar, salt, and apples; cook, stirring occasionally, until apples are slightly caramelized, about 20 minutes.

Pull pan from heat and add armagnac; carefully ignite with a match.

Return to heat and cook until flames subside and liquid is reduced by half, about 3 minutes; let cool.

Preheat the oven to 425°F (220°C).

Grease a 10" springform pan with butter; set aside.

Melt the remaining butter in a 1-qt. saucepan; keep warm.

Lay 1 sheet phyllo on a work surface; brush with some melted butter and sprinkle with 1 tsp. granulated sugar.

Fit into prepared pan, allowing corners of dough to hang over edges.

Repeat using 3 more sheets phyllo, laying each sheet at a 45° turn from the last.

Spread apples over dough in an even layer.

Cut remaining sheets of phyllo in quarters.

Working with 1 piece at a time, pinch at the center and flip over, so that the corners are pulled together and facing up; place over top of tart.

Repeat with remaining pieces phyllo until top of tart is covered.

Drizzle with remaining melted butter; bake until golden and crisp, about 30 minutes.

Let tart cool completely in the pan, then unmold and transfer to a serving platter.