Make the bread: Whisk together ⅓ cup flour and ½ cup water in a 1-qt. saucepan over medium-high heat and cook, stirring often, until thickened into a paste. Transfer to a small bowl and refrigerate until chilled. Combine 1 cup flour, ¼ tsp. yeast, and ⅓ cup water in a medium bowl, to form a wet dough. Cover with plastic wrap and let starter sit at room temperature overnight.
Whisk together remaining flour and yeast in bowl of a stand mixer fitted with a dough hook; set aside. In a bowl, whisk together sugar, condensed milk, salt, and ½ cup water heated to 120°, until sugar dissolves. Add to bowl of flour, along with chilled flour paste and room-temperature starter; stir to form a shaggy dough; turn mixer to medium speed and knead for 6–8 minutes. With the mixer running, add butter, 1 piece at a time, until incorporated; then add oil and mix. Divide dough into 3 equal pieces; add a different food coloring to each piece; knead until evenly tinted. Transfer dough pieces to 3 separate bowls, cover with plastic wrap, and let sit until doubled in size, about 2 hours.
Transfer dough pieces to a work surface and roll into 12"-long ropes; braid ropes together and tuck ends under, to form a loaf. Transfer to a greased 8½" × 5" × 2" loaf pan and cover loosely with plastic wrap; let sit for 1½ hours. Heat oven to 400°. Uncover loaf; bake until golden brown and an instant-read thermometer inserted into loaf reads 185°, 40 to 45 minutes. Let cool 30 minutes, remove from pan, and cool completely. To serve, slice bread into ½"-thick slices. Place scoops of ice cream on every slice.