Preheat oven to 375 degrees F (190 degrees C). Grease a 9x13-inch glass baking pan.
Heat almond milk over medium heat until frothy, about 5 minutes. Remove from heat and stir in 2 tablespoons vegetable oil. Cool until milk is lukewarm.
Stir 1/4 cup plus 3 tablespoons warm water and ground flax together in a measuring cup.
Combine 2 1/4 cups flour, white sugar, yeast, and salt in a large bowl. Add milk mixture and water mixture; mix well. Stir in remaining flour 1/2 cup at a time until dough comes together.
Turn dough out onto a lightly floured surface and knead until smooth and springy, 5 to 7 minutes. Return dough to the bowl and cover with a damp cloth. Place on top of the oven and let rest for 10 to 20 minutes.
Mix brown sugar, vanilla extract, almond extract, cinnamon, nutmeg, and cardamom together in a bowl. Stir in 3 tablespoons vegetable oil.
Return dough to the floured surface and roll into a large rectangle. Spread the brown sugar mixture onto the dough. Sprinkle pecans on top. Roll the dough along the longer side of the rectangle. Use a length of thread to cut the dough into 1 1/2-inch segments. Place into the prepared baking pan swirl-side up.
Cover baking pan with a damp cloth and place on top of the oven. Let rise for 30 to 40 minutes. Remove cloth.
Bake until rolls are golden brown, about 30 minutes.