Pat chicken thighs dry, and season both sides with 1/2 tsp. salt and 1/4 tsp. pepper. Place a medium non-stick pan over medium-high heat. Add 2 tsp. olive oil and chicken thighs to hot pan. Cook until browned on both sides, 3-5 minutes per side. Transfer chicken to a plate. Chicken will finish cooking in a later step. Wipe pan clean and reserve. While chicken sears, prepare ingredients.

Cut mushrooms into 1/4" slices. Peel, trim, and cut carrot into 1/4" dice. Trim woody ends off asparagus and cut into 1/2" lengths. Stem and coarsely chop thyme.

Return pan used to sear chicken to medium-high heat. Add 2 tsp. olive oil and mushrooms to hot pan. Cook, stirring occasionally, until lightly browned, 2-3 minutes. Reduce heat to medium and add carrots and 1/4 tsp. salt. Cook, stirring occasionally, until carrots begin to soften, 3-4 minutes. Add 2 Tbsp. flour (reserve remaining for dusting hands) and stir, 1 minute.

Add white cooking wine, chicken demi-glace, and 1 cup water and raise heat to high. Bring to a boil and cook until thick enough to coat the back of a spoon, 3-4 minutes. Slice chicken into 1/4" slices. Remove pan from burner and stir in thyme, sliced chicken and any accumulated juices, and asparagus. Scoop stew into prepared casserole dishes.

Dust hands with reserved flour. Remove puff pastry squares from refrigerator. Use the back of a small knife to draw diagonal lines on puff pastry square, 1/2" apart. Draw another set of lines in opposite direction to make a crosshatch pattern. Gently stretch puff pastry squares over casserole dish edges and carefully press dough into side to seal. Place on a baking sheet and bake until well-browned and chicken reaches a minimum internal temperature of 165 degrees, 10-12 minutes.

Remove from baking sheet and serve pot pie directly from casserole dishes.