MAKE THE CARAMELIZED ONIONS:
Heat butter and olive oil in large skillet over medium heat. Add sliced onions and salt. Cook stirring occasionally for 30-35 minutes until deeply golden and caramelized. Patience is key don't rush this step! If onions start to stick or burn reduce heat and add splash of water. In final 5 minutes add brown sugar, balsamic vinegar, pepper, and thyme. Cook until liquid evaporates and onions are jammy and sweet. Cool completely then chop finely if onions are in large pieces.

MAKE THE SAVORY BISCUIT DOUGH:
In food processor pulse flour, baking powder, salt, pepper, and thyme. Add cold butter cubes and Parmesan. Pulse until mixture resembles coarse breadcrumbs. Beat egg with 2 tablespoons cold water. Add to flour mixture and pulse just until dough comes together adding more water if needed. Turn out onto lightly floured surface, knead gently 2-3 times, then shape into disk. Wrap in plastic wrap and refrigerate at least 1 hour.

ROLL AND CUT THE BISCUITS ADDRESSING MARTHA'S UNEVEN BAKING:
Preheat oven to 350°F (175°C). Line two baking sheets with parchment paper. Roll out chilled dough on lightly floured surface to exactly 1/8-inch thickness. Consistent thickness is critical for even baking. Use 2-inch round cutter or desired shape to cut biscuits. Re-roll scraps once being gentle to maintain tenderness. Place biscuits on prepared baking sheets spacing 1 inch apart. Docking Martha's custom tool: Using fork gently prick each biscuit 3-4 times to prevent puffing and ensure even baking. Brush lightly with egg wash and sprinkle with flaky sea salt.

BAKE:
For even baking: Bake one sheet at a time in center of oven for 12-14 minutes until edges are lightly golden. Rotate sheet halfway through baking for even color. Cool on baking sheet 5 minutes then transfer to wire rack to cool completely. Critical: Biscuits must be completely cool before filling or goat cheese will melt.

MAKE THE GOAT CHEESE FILLING:
In bowl beat softened goat cheese with heavy cream until smooth and spreadable. Fold in chopped chives, lemon zest, pepper, and pinch of salt. Taste and adjust seasoning the filling should be tangy and herbaceous.

ASSEMBLE THE SANDWICH BISCUITS:
Spread about 1 tablespoon caramelized onions on flat side of half the biscuits. Spread or pipe about 1 tablespoon goat cheese filling on flat side of remaining biscuits. Gently press goat cheese biscuit and onion biscuit together to create sandwiches. Martha's ambitious goal: She made 36 sandwiches (72 individual biscuits) this recipe makes more manageable 18 sandwiches for home bakers!