Mix wet ingredients in food processor first, then add flour and mix until well combined.
Spray crepe pan with cooking spray (spread with a silicone brush to cover entire pan). Pour 1 ladle of batter (about 1/4 cup) into the pan and either use a crepe spreader or swirl the pan to coat to the edges. Cook until the top of the crepe is no longer wet and the bottom has turned light brown, 1 to 2 minutes. Run a spatula around the edge of the crepe to loosen it from the pan. Flip and cook until the other side has turned light brown, about another minute.
Stack crepes on a pan and keep warm.
Filling and Sauce
In a large skillet add the mushrooms, then turn on medium high heat. Let the moisture render and cook until browned, about 4 minutes. Flip and continue cooking until tender and browned, stirring occasionally, about 5 minutes. Move the mushrooms to the sides of the pan.
Add the pat of butter and shallot in the middle. Stir constantly until fragrant, about 1 minute. Stir in the cream, chicken broth, dijon mustard, garlic powder, salt, and white pepper and simmer, stirring occasionally to check the consistency, until thickened, 5 to 8 minutes. Add the Swiss cheese one handful at a time and stir until smooth and melted.
Measure out 3/4 cup of the mushroom sauce and set aside in a small bowl for garnishing later.
Fold in the chicken and cook until heated through, about 1 minute.
Add the baby spinach to the sauce remaining in the pan and cook, stirring often, until the spinach is wilted, approximately 2 minutes.
Spoon 1/4 cup of the chicken mixture onto each crepe and either roll up or fold into a triangle. Serve immediately, topped with reserved mushroom sauce.