In the bowl of a stand mixer, whisk together 1 cup (125 grams) of flour, granulated sugar, salt, and yeast by hand.
In a small saucepan, heat ¼ cup (57 grams) of butter, ¼ cup (60 grams) of water, whole milk, and vanilla over medium-low heat until the butter is melted and the mixture registers between 120°F (49°C) to 130°F (54°C).
Add the warm butter mixture to the flour mixture in the stand mixer.
Using the paddle attachment, beat at medium-low speed until all ingredients are combined.
Add 1 egg (50 grams) and mix until fully incorporated.
With the mixer on low speed, gradually add 1½ cups (188 grams) of flour, mixing just until combined.
Switch to the dough hook attachment and beat at medium-low speed until a soft, somewhat sticky dough forms.
Turn out the dough onto a very lightly floured surface and shape it into a smooth round.
Lightly oil a large bowl, place the dough in it, and cover.
Let it rise in a warm, draft-free place (around 75°F/24°C) until it doubles in size.
Once the dough has risen, turn it out onto a lightly floured surface and gently press into an 8x6-inch rectangle.
Loosely wrap it in plastic wrap and refrigerate for at least 1 hour or up to overnight.
Preheat your oven to 375°F (190°C).
Spray an 8½x4½-inch loaf pan with cooking spray and line it with parchment paper.
In a small bowl, mix together the remaining butter, brown sugar, cinnamon, and the remaining salt.
On a lightly floured surface, roll the dough into a 13x10-inch rectangle.
Spread the butter mixture onto the dough, leaving a ½-inch border.
Sprinkle the pecans on top and press lightly to adhere.
Starting from the long side opposite the border, roll up the dough jelly roll style and pinch the seam to seal.
Gently stretch the log to about 14 inches long.
Using a sharp serrated knife, cut the log in half lengthwise.
Turn the halves cut side up and place them in an “X” pattern.
Twist the top half twice and pinch the ends together, tucking them underneath.
Repeat this with the bottom half of the “X.”
Place the twisted dough into the prepared pan, ensuring the cut sides are up and ends are tucked in.
Cover and let rise until puffed, about 30 minutes to 1 hour.
In a small bowl, whisk together the remaining egg and the remaining water.
Brush the egg wash on top of the dough.
Bake until golden brown and an instant-read thermometer reads 190°F (88°C).
Let the babka cool in the pan for about 10 minutes, then remove and cool completely on a wire rack.