Heat 1 Tbsp of butter on low in a saute pan. Add the sliced shallots, a pinch of salt. Let cook, stirring occasionally until caramelized- about 15 minutes. 

Preheat the oven to 500 degrees. Let the beef come to room temperature. Slather with the last 3 Tbsp of butter, 1 Tbsp of kosher salt and 1 Tsp of the black pepper. Bake in the oven for EXACTLY 25 minutes for medium rare. Take it out of the oven and immediately cover with foil and let it sit. When it has cooled and rested, slice thin.  

Slice the baguette and drizzle lightly with olive oil and kosher salt. Toast under the broiler until golden brown, then flip the slices and toast the other side. 

Mix the mayo, fig jam, fresh thyme and salt and pepper in a bowl. 

Assemble the crostinis. Smear about a tablespoon of the mayo-fix mixture on the baguette slice. Top with a slice of the tenderloin and finish off with some caramelized shallots. The steps can all be made ahead of time and then assembled when your guests arrive.